{"comments":[{"comment":{"id":9604235,"creator_id":1118197,"post_id":10328929,"content":"if 0.333... = 1/3 then 0.999... = 3/3 = 1","removed":false,"published":"2024-07-02T07:50:21.500876Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074484","local":false,"path":"0.9604235","distinguished":false,"language_id":37},"creator":{"id":1118197,"name":"iie","display_name":"iie [they/them, he/him]","banned":false,"published":"2020-07-30T23:45:38.914758Z","updated":"2023-02-27T07:29:48.276744Z","actor_id":"https://hexbear.net/u/iie","bio":"I go by \"test\" on live.hexbear.net, or \"tset\" or \"tst\" or some other variant when I'm not logged in.\n\nWe watch movies on the weekends and sometimes also hang out during the week, you should drop by.","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604235,"score":13,"upvotes":13,"downvotes":0,"published":"2024-07-02T07:50:21.500876Z","child_count":1},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604899,"creator_id":1102982,"post_id":10328929,"content":"The thing that really got 0.999...=1 to click on my mind is the fact that you can't find a number *between* 0.999... and 1. You might think \"Just put something at the end of 0.999...,\" but there is no end to 0.999....","removed":false,"published":"2024-07-02T09:20:08.449470Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074596","local":false,"path":"0.9604899","distinguished":false,"language_id":37},"creator":{"id":1102982,"name":"TC_209","display_name":"TC_209 [he/him, pup/pup's]","avatar":"https://hexbear.net/pictrs/image/2881f88c-f908-4401-bbad-b7d75613d4bf.jpeg","banned":false,"published":"2020-08-14T04:22:32.641358Z","updated":"2023-03-03T07:54:27.449245Z","actor_id":"https://hexbear.net/u/TC_209","bio":"Open to DMs if you have something to say but don't want to make it a reply to a post.","local":false,"banner":"https://hexbear.net/pictrs/image/70e2e811-ec02-49b9-aea0-212571c84b2e.webp","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604899,"score":10,"upvotes":10,"downvotes":0,"published":"2024-07-02T09:20:08.449470Z","child_count":8},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604236,"creator_id":1161439,"post_id":10328929,"content":">and wants to know why 0.999... = 1\n\n\\begin{align}\n0.999.... &= 9\\cdot(0.1+0.01+0.001+... ) \\\\\n&= 9\\cdot( 0.1^1 + 0.1^2 + 0.1^3 + ... ) \\\\\n&= 9\\cdot(\\sum\\limits_{k=1}^\\infty ( \\frac{1}{10^k} ) ) \\\\\n&= 9\\cdot(\\sum\\limits_{k=0}^\\infty ( \\frac{1}{10^{(k+1)}}))\\\\\n&= 9\\cdot(\\sum\\limits_{k=0}^\\infty \\frac{1}{10}*(\\frac{1}{10^k})) \\\\\n&= \\frac{9}{10}\\cdot (\\sum\\limits_{k=0}^\\infty (\\frac{1}{10^k})) \\\\\n&= \\frac{9}{10}\\cdot \\frac{1}{(1-(\\frac{1}{10}))}\\\\\n&= \\frac{9}{10}\\cdot \\frac{10}{9} = 1\n\\end{align}\n\nThe crux rests on a handy result on from calculus: [the sum of an infinite geometric series](https://www.youtube.com/watch?v=wqnpSzEzq1w) looks likes s = 1/(1-r), when s = \\sum\\limits_k=0^inf r^k, and |r| < 1.\n\nSorry for the latex. When will hexbear render latex? This is a bit more readable:\n\n\n(aesthetic edit for our big beautiful complex analysts)\n\n![](https://hexbear.net/pictrs/image/fb5fea59-ae7d-4394-8a44-49924038e814.png)","removed":false,"published":"2024-07-02T07:50:37.734520Z","updated":"2024-07-02T16:53:57.772839Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074485","local":false,"path":"0.9604236","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604236,"score":9,"upvotes":9,"downvotes":0,"published":"2024-07-02T07:50:37.734520Z","child_count":13},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604302,"creator_id":1172154,"post_id":10328929,"content":"0.999..=1 because it's funny that people get so mad about it","removed":false,"published":"2024-07-02T07:59:48.074581Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074496","local":false,"path":"0.9604302","distinguished":false,"language_id":37},"creator":{"id":1172154,"name":"NephewAlphaBravo","display_name":"NephewAlphaBravo [he/him]","banned":false,"published":"2020-08-14T01:44:20.763966Z","updated":"2021-03-15T03:55:52.286716Z","actor_id":"https://hexbear.net/u/NephewAlphaBravo","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604302,"score":9,"upvotes":9,"downvotes":0,"published":"2024-07-02T07:59:48.074581Z","child_count":2},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9607430,"creator_id":2296592,"post_id":10328929,"content":"so, start with:\n\nx = 0.999...\n\nnow multiply each side by 10\n\n10x = 9.9999....\n\nnow we subtract x from the left side, and 0.999... from the right, which is fine because they are equal:\n\n9x = 9\n\nand from there it should be fairly obvious that x is also equal to 1, which means 0.999... is also equal to 1","removed":false,"published":"2024-07-02T13:06:19.019658Z","deleted":false,"ap_id":"https://lemmygrad.ml/comment/4571512","local":false,"path":"0.9607430","distinguished":false,"language_id":37},"creator":{"id":2296592,"name":"bestmiaou","avatar":"https://lemmygrad.ml/pictrs/image/8493f6c4-6e31-491e-a6c1-02ff34d72478.png","banned":false,"published":"2022-03-05T02:29:25.589571Z","updated":"2022-05-27T22:00:47.185092Z","actor_id":"https://lemmygrad.ml/u/bestmiaou","bio":"pronouns: she/her","local":false,"deleted":false,"bot_account":false,"instance_id":36},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9607430,"score":8,"upvotes":8,"downvotes":0,"published":"2024-07-02T13:06:19.019658Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604258,"creator_id":6177520,"post_id":10328929,"content":"He's hungry \n![cri](https://www.hexbear.net/pictrs/image/a56152fe-02c6-41e3-b2f3-865b6c564422.png \"emoji cri\") \nFeed him","removed":false,"published":"2024-07-02T07:53:25.812889Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074489","local":false,"path":"0.9604258","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604258,"score":7,"upvotes":7,"downvotes":0,"published":"2024-07-02T07:53:25.812889Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9605119,"creator_id":6177520,"post_id":10328929,"content":"I will be posting another another proof, one using the assumption that 0.999... is the sum of the series 9/10+9/100+9/1000+... (and that 1 is the sum of the series 1+0+0+0+...), unless somebody else comes here and feeds it to the little guy first.","removed":false,"published":"2024-07-02T09:45:31.911784Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074633","local":false,"path":"0.9605119","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9605119,"score":6,"upvotes":6,"downvotes":0,"published":"2024-07-02T09:45:31.911784Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604431,"creator_id":6177520,"post_id":10328929,"content":"I suppose I will post one myself, as I do not expect anybody else to have that one in mind.\n\nThe decimals '0.999...' and '1' refer to the real numbers that are equivalence classes of Cauchy sequences of rational numbers (0.9, 0.99, 0.999,...) and (1, 1, 1,...) with respect to the relation R: (aRb) <=> (lim(a_n-b_n) as n->inf, where a_n and b_n are the nth elements of sequences a and b, respectively).\n\nFor a = (1, 1, 1,...) and b = (0.9, 0.99, 0.999,...) we have lim(a_n-b_n) as n->inf = lim(1-sum(9/10^k) for k from 1 to n) as n->inf = lim(1/10^n) as n->inf = 0. That means that (1, 1, 1,...)R(0.9, 0.99, 0.999,...), i.e. that these sequences belong to the same equivalence class of Cauchy sequences of rational numbers with respect to R. In other words, the decimals '0.999...' and '1' refer to the same real number. QED.","removed":false,"published":"2024-07-02T08:14:51.175297Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074513","local":false,"path":"0.9604431","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604431,"score":6,"upvotes":6,"downvotes":0,"published":"2024-07-02T08:14:51.175297Z","child_count":9},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9617721,"creator_id":1110117,"post_id":10328929,"content":"(I haven't done real math since forever)\n\n##### Part 1: Proving If x,y are distinct positive real numbers and x < y, there is a z in **R** such that x < z < y.\n\nLet y = x + e, where e is in **R**. e is also positive because:\n\nx < y \nx < x + e \n0 < e\n\nx < z < y becomes x < z < x + e. We can then choose z = x + 0.5e so that\n\nx < z < y \nx < x + 0.5e < x + e \n0 < 0.5e < e \n0.5e - 0.5e < 0.5e + 0 < 0.5e + 0.5e \n-0.5e < 0 < 0.5e\n\nSince we've shown earlier that e > 0, -0.5e < 0 < 0.5e is true no matter what e.\n\n##### Part 2: Decimal Notation \n\nDecimal notation is written as the sequence a~m~a~m-1~...a~1~a~0~.b~1~b~2~...b~n~, where a,b are in the set {0,1,2,3,4,5,6,7,8,9}. The sequence a~m~a~m-1~...a~1~a~0~.b~1~b~2~...b~n~ itself is of the number a~m~10^m^ + a~m-1~10^m-1^ + ... + a~0~10^0^ + b~1~10^-1^ + b~2~10^-2^ + ... + b~n~10^-n^.\n\n##### Part 3: The actual proof through proof by contradiction\n\nLet x = 0.999... and y = 1. This means there exists a z that is between 0.999... and 1. So let's construct z = a~0~.b~1~b~2~...b~n~. a~0~ has to be either a 0 or 1 and since there is no number smaller than 1 with 1 as its first digit, a~0~=0. z = 0.b~1~b~2~...b~n~\n\n0.999... = 9\\*10^-1^ + 9\\*10^-2^ + 9\\*10^-3^ + ... while z = b~1~10^-1^ + b~2~10^-2^ + ... + b~n~10^-n^. If 0.999... < z, then \n\n0.999... - z < 0 \n9\\*10^-1^ + 9\\*10^-2^ + 9\\*10^-3^ + ... - (b~1~10^-1^ + b~2~10^-2^ + ... + b~n~10^-n^) < 0 \n(9\\*10^-1^ - b~1~10^-1^) + (9\\*10^-2^ - b~2~10^-2^) + (9\\*10^-3^ - b~3~10^-3^) + ... < 0 \n(9 - b~1~)\\*10^-1^ + (9 - b~2~)\\*10^-2^ + (9 - b~3~)\\*10^-3^ + ... < 0 \n\nBut as we've established in Part 2, b~1~, b~2~, b~3~, etc have to be from the set {0,1,2,3,4,5,6,7,8,9}, meaning (9 - b~n~) > 0 for any n. Therefore, (9 - b~1~)\\*10^-1^ + (9 - b~2~)\\*10^-2^ + (9 - b~3~)\\*10^-3^ + ... cannot be less than 0.\n\nThe theorem requires for x and y to be distinct positive real numbers and x < y, and since 0.999... and 1 can be trivially shown to be positive real numbers, this means that 0.999... and 1 are *not* distinct. In other words, 0.999... = 1.","removed":false,"published":"2024-07-03T02:12:23.696847Z","deleted":false,"ap_id":"https://hexbear.net/comment/5077646","local":false,"path":"0.9617721","distinguished":false,"language_id":37},"creator":{"id":1110117,"name":"AssortedBiscuits","display_name":"AssortedBiscuits [they/them]","avatar":"https://hexbear.net/pictrs/image/8bdaa986-a60a-4798-a373-b4f48ee3f026.jpeg","banned":false,"published":"2022-05-22T00:04:03.918368Z","updated":"2022-10-12T21:15:15.918206Z","actor_id":"https://hexbear.net/u/AssortedBiscuits","bio":"mfw you still use Windows in ~~2023~~ ~~2024~~ 2025\n![](https://hexbear.net/api/v3/image_proxy?url=https%3A%2F%2Fwww.hexbear.net%2Fpictrs%2Fimage%2Fff5e1425-1fcb-4155-92e0-83b987c61399.jpeg)","local":false,"banner":"https://hexbear.net/pictrs/image/0ea97295-22bc-4f8d-bae7-398e51cf9726.gif","deleted":false,"bot_account":false,"ban_expires":"2024-12-04T05:42:22Z","instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9617721,"score":3,"upvotes":3,"downvotes":0,"published":"2024-07-03T02:12:23.696847Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9608188,"creator_id":1122552,"post_id":10328929,"content":"lots of good proofs in this thread but this one gets points for being by *FAR* the shortest while still being completely based on intuition and makes complete sense with an average high school mathematics background","removed":false,"published":"2024-07-02T13:53:44.851758Z","deleted":false,"ap_id":"https://hexbear.net/comment/5075138","local":false,"path":"0.9604235.9608188","distinguished":false,"language_id":37},"creator":{"id":1122552,"name":"aaro","display_name":"aaro [they/them, she/her]","banned":false,"published":"2020-10-22T14:47:58.482124Z","updated":"2022-12-28T21:30:12.438690Z","actor_id":"https://hexbear.net/u/aaro","bio":"touch grass, eat ass, abolish class ![heart-sickle](https://www.hexbear.net/pictrs/image/14c2707b-51cb-44ed-be2f-f262d088e219.png \"emoji heart-sickle\") ","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9608188,"score":10,"upvotes":10,"downvotes":0,"published":"2024-07-02T13:53:44.851758Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604993,"creator_id":2330698,"post_id":10328929,"content":"Never thought of employing the geometric series for this, that’s clever. ","removed":false,"published":"2024-07-02T09:31:42.934398Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074615","local":false,"path":"0.9604236.9604993","distinguished":false,"language_id":37},"creator":{"id":2330698,"name":"reaper_cushions","display_name":"reaper_cushions [he/him, comrade/them]","banned":false,"published":"2020-11-07T01:37:22.299812Z","updated":"2021-01-19T15:38:36.106745Z","actor_id":"https://hexbear.net/u/reaper_cushions","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604993,"score":8,"upvotes":8,"downvotes":0,"published":"2024-07-02T09:31:42.934398Z","child_count":1},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9606691,"creator_id":27581,"post_id":10328929,"content":"Using i as an index 🤮","removed":false,"published":"2024-07-02T12:14:24.516083Z","deleted":false,"ap_id":"https://lemmygrad.ml/comment/4571294","local":false,"path":"0.9604236.9606691","distinguished":false,"language_id":37},"creator":{"id":27581,"name":"Xavienth","display_name":"Xavienth","banned":false,"published":"2020-07-18T05:53:22.352455Z","actor_id":"https://lemmygrad.ml/u/Xavienth","local":false,"deleted":false,"bot_account":false,"instance_id":36},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9606691,"score":3,"upvotes":3,"downvotes":0,"published":"2024-07-02T12:14:24.516083Z","child_count":6},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604734,"creator_id":7262644,"post_id":10328929,"content":"","removed":false,"published":"2024-07-02T08:56:03.463262Z","updated":"2024-08-31T10:32:28.464051Z","deleted":true,"ap_id":"https://hexbear.net/comment/5074566","local":false,"path":"0.9604236.9604734","distinguished":false,"language_id":37},"creator":{"id":7262644,"name":"TheWurstman","banned":false,"published":"2024-03-11T18:46:28.553762Z","updated":"2024-08-31T10:32:36.357078Z","actor_id":"https://hexbear.net/u/TheWurstman","local":false,"deleted":true,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604734,"score":2,"upvotes":2,"downvotes":0,"published":"2024-07-02T08:56:03.463262Z","child_count":3},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604818,"creator_id":6177520,"post_id":10328929,"content":"A real number can't be approaching anything. It is not a function or any other sort of object that can be said to be approaching anything.","removed":false,"published":"2024-07-02T09:08:08.904017Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074584","local":false,"path":"0.9604236.9604734.9604818","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604818,"score":8,"upvotes":8,"downvotes":0,"published":"2024-07-02T09:08:08.904017Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604759,"creator_id":1161439,"post_id":10328929,"content":"No. 0.99 is 0.9+0.09. \nThe proof I gave shows that 0.99999999999999999999999999999999999(...) is equal to 1.","removed":false,"published":"2024-07-02T09:00:17.583622Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074571","local":false,"path":"0.9604236.9604734.9604759","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604759,"score":5,"upvotes":5,"downvotes":0,"published":"2024-07-02T09:00:17.583622Z","child_count":1},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9605035,"creator_id":1161439,"post_id":10328929,"content":"\\^_^ thank you!","removed":false,"published":"2024-07-02T09:36:52.325632Z","updated":"2024-07-02T09:36:59.880276Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074623","local":false,"path":"0.9604236.9604993.9605035","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9605035,"score":5,"upvotes":5,"downvotes":0,"published":"2024-07-02T09:36:52.325632Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9608170,"creator_id":2364755,"post_id":10328929,"content":"? Thats pretty standard though right?","removed":false,"published":"2024-07-02T13:52:46.013339Z","deleted":false,"ap_id":"https://hexbear.net/comment/5075133","local":false,"path":"0.9604236.9606691.9608170","distinguished":false,"language_id":37},"creator":{"id":2364755,"name":"mathemachristian","display_name":"mathemachristian [he/him]","avatar":"https://chapo.chat/pictrs/image/75b8469d-1319-444c-81be-14e39ead5a6d.png","banned":false,"published":"2023-09-15T10:51:07.444481Z","actor_id":"https://hexbear.net/u/mathemachristian","local":false,"banner":"https://chapo.chat/pictrs/image/d4fed5d7-4ff9-443b-93b1-32daee6827e6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9608170,"score":5,"upvotes":5,"downvotes":0,"published":"2024-07-02T13:52:46.013339Z","child_count":4},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9611404,"creator_id":1161439,"post_id":10328929,"content":"Your feedback is valid and I apologize for rendering such an ugly proof","removed":false,"published":"2024-07-02T16:44:59.697979Z","deleted":false,"ap_id":"https://hexbear.net/comment/5075727","local":false,"path":"0.9604236.9606691.9611404","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9611404,"score":1,"upvotes":1,"downvotes":0,"published":"2024-07-02T16:44:59.697979Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604314,"creator_id":6177520,"post_id":10328929,"content":"I agree, but the little guy does need rigour, or he will starve under crapitalism.","removed":false,"published":"2024-07-02T08:00:58.871342Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074499","local":false,"path":"0.9604302.9604314","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604314,"score":8,"upvotes":8,"downvotes":0,"published":"2024-07-02T08:00:58.871342Z","child_count":1},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604335,"creator_id":1172154,"post_id":10328929,"content":"I didn't have any bread to bring him but I had plenty of roses","removed":false,"published":"2024-07-02T08:03:17.973428Z","updated":"2024-07-02T08:04:02.429655Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074500","local":false,"path":"0.9604302.9604314.9604335","distinguished":false,"language_id":37},"creator":{"id":1172154,"name":"NephewAlphaBravo","display_name":"NephewAlphaBravo [he/him]","banned":false,"published":"2020-08-14T01:44:20.763966Z","updated":"2021-03-15T03:55:52.286716Z","actor_id":"https://hexbear.net/u/NephewAlphaBravo","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604335,"score":6,"upvotes":6,"downvotes":0,"published":"2024-07-02T08:03:17.973428Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604848,"creator_id":1161439,"post_id":10328929,"content":"I like how compact this one is ;)","removed":false,"published":"2024-07-02T09:12:19.985627Z","updated":"2024-07-02T09:12:35.882536Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074588","local":false,"path":"0.9604431.9604848","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604848,"score":4,"upvotes":4,"downvotes":0,"published":"2024-07-02T09:12:19.985627Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604725,"creator_id":7262644,"post_id":10328929,"content":"","removed":false,"published":"2024-07-02T08:55:12.659599Z","updated":"2024-08-31T10:32:28.464051Z","deleted":true,"ap_id":"https://hexbear.net/comment/5074565","local":false,"path":"0.9604431.9604725","distinguished":false,"language_id":37},"creator":{"id":7262644,"name":"TheWurstman","banned":false,"published":"2024-03-11T18:46:28.553762Z","updated":"2024-08-31T10:32:36.357078Z","actor_id":"https://hexbear.net/u/TheWurstman","local":false,"deleted":true,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604725,"score":2,"upvotes":2,"downvotes":0,"published":"2024-07-02T08:55:12.659599Z","child_count":5},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9615668,"creator_id":1105065,"post_id":10328929,"content":"Had a little trouble following this as plain text, so I wrote it up in LaTeX (it'll be a bit small if you try to read it inline--you'll probably want to tap to enlarge on mobile or open the image in a new tab/[click this direct link to the image](https://hexbear.net/pictrs/image/2fe10eb2-1a2b-4e6c-be33-b4a55fb43dc0.png)):\n\n![](https://hexbear.net/pictrs/image/2fe10eb2-1a2b-4e6c-be33-b4a55fb43dc0.png)\n\nI tried to hew as closely to your notation as I could, but let me know if you spot any errors!","removed":false,"published":"2024-07-02T17:37:36.447603Z","deleted":false,"ap_id":"https://hexbear.net/comment/5075943","local":false,"path":"0.9604431.9615668","distinguished":false,"language_id":37},"creator":{"id":1105065,"name":"AernaLingus","display_name":"AernaLingus [any]","banned":false,"published":"2022-05-06T08:55:37.859321Z","actor_id":"https://hexbear.net/u/AernaLingus","local":false,"deleted":false,"matrix_user_id":"@aernalingus:catgirl.cloud","bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9615668,"score":0,"upvotes":0,"downvotes":0,"published":"2024-07-02T17:37:36.447603Z","child_count":1},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604807,"creator_id":1161439,"post_id":10328929,"content":"Not quite. The wording \"equivalence classes of ... with respect to the relation R: aRb <==> lim( a_n - b_n) as ->inf\" is key. \n\nhttps://en.wikipedia.org/wiki/Equivalence_class\n\nloosely, an equivalence relation is a relation between things in a set that behaves the way we want an equal sign to\n\nFor an element in a set, a, the equivalence class of a is the set of all things in the larger set that are equivalent to a. ","removed":false,"published":"2024-07-02T09:06:33.310794Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074581","local":false,"path":"0.9604431.9604725.9604807","distinguished":false,"language_id":37},"creator":{"id":1161439,"name":"dat_math","display_name":"dat_math [they/them]","avatar":"https://hexbear.net/pictrs/image/cdf15731-8894-406e-81d2-2cf3409f759f.png","banned":false,"published":"2021-07-09T07:42:24.589537Z","actor_id":"https://hexbear.net/u/dat_math","local":false,"banner":"https://hexbear.net/pictrs/image/1573be35-e71d-4f62-be4b-462679a7bfd6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604807,"score":6,"upvotes":6,"downvotes":0,"published":"2024-07-02T09:06:33.310794Z","child_count":3},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9604801,"creator_id":6177520,"post_id":10328929,"content":"So, under the relevant construction of the space of real numbers, every real number is an **equivalence class** of Cauchy sequences of rational numbers with respect to the relation R outlined in my comment. In other words, under this definition, a real number is an equivalence class that includes all such sequences that for every pair of them the relation R holds (and R is, indeed, an equivalence relation - it is reflexive, symmetric, and transitive, - that is not hard to prove).\n\nWe prove that, for the sequences (1, 1, 1,...) and (0.9, 0.99, 0.999,...), the relation R holds, which means that they are both in the same equivalence class of those sequences.\n\nThe decimals '1' and '0.999...', under the relevant definition, refer to numbers that are equivalence classes that include the aforementioned sequences as their elements. However, as we have proven, the sequences both belong to the same equivalence class, meaning that the decimals '1' and '0.999...' refer to the same equivalence class of Cauchy sequences of rational numbers with respect to R, i.e. they refer to the same real number, i.e. 0.999... = 1.","removed":false,"published":"2024-07-02T09:05:22.083863Z","updated":"2024-07-02T09:10:26.034891Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074579","local":false,"path":"0.9604431.9604725.9604801","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9604801,"score":5,"upvotes":5,"downvotes":0,"published":"2024-07-02T09:05:22.083863Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9615669,"creator_id":6177520,"post_id":10328929,"content":"Thank you.\n\nAnd no, I do not notice any errors.\n\nI should re-learn Lateχ.","removed":false,"published":"2024-07-02T19:05:16.979263Z","updated":"2024-07-02T19:06:02.933794Z","deleted":false,"ap_id":"https://hexbear.net/comment/5076292","local":false,"path":"0.9604431.9615668.9615669","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9615669,"score":1,"upvotes":1,"downvotes":0,"published":"2024-07-02T19:05:16.979263Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9605576,"creator_id":2364755,"post_id":10328929,"content":"","removed":false,"published":"2024-07-02T10:37:31.742590Z","deleted":true,"ap_id":"https://hexbear.net/comment/5074696","local":false,"path":"0.9604899.9605576","distinguished":false,"language_id":37},"creator":{"id":2364755,"name":"mathemachristian","display_name":"mathemachristian [he/him]","avatar":"https://chapo.chat/pictrs/image/75b8469d-1319-444c-81be-14e39ead5a6d.png","banned":false,"published":"2023-09-15T10:51:07.444481Z","actor_id":"https://hexbear.net/u/mathemachristian","local":false,"banner":"https://chapo.chat/pictrs/image/d4fed5d7-4ff9-443b-93b1-32daee6827e6.png","deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9605576,"score":2,"upvotes":2,"downvotes":0,"published":"2024-07-02T10:37:31.742590Z","child_count":6},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9611826,"creator_id":1099981,"post_id":10328929,"content":"Yay, Dedekind cuts are finally useful!","removed":false,"published":"2024-07-02T17:07:34.780886Z","deleted":false,"ap_id":"https://hexbear.net/comment/5075825","local":false,"path":"0.9604899.9611826","distinguished":false,"language_id":37},"creator":{"id":1099981,"name":"Collatz_problem","display_name":"Collatz_problem [comrade/them]","banned":false,"published":"2020-11-13T17:48:56.835079Z","actor_id":"https://hexbear.net/u/Collatz_problem","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9611826,"score":2,"upvotes":2,"downvotes":0,"published":"2024-07-02T17:07:34.780886Z","child_count":0},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false},{"comment":{"id":9605655,"creator_id":6177520,"post_id":10328929,"content":"> Yeah that's something that people have to get used to in maths, if the limit of a sequence exists we can just pretend to have \"reached infinity\" and work with like any number\n\nI'm not sure what you are trying to say here, and I have a background in math. I think this is just going to confuse lay people.","removed":false,"published":"2024-07-02T10:48:10.794677Z","deleted":false,"ap_id":"https://hexbear.net/comment/5074715","local":false,"path":"0.9604899.9605576.9605655","distinguished":false,"language_id":37},"creator":{"id":6177520,"name":"Tomorrow_Farewell","display_name":"Tomorrow_Farewell [any, they/them]","avatar":"https://hexbear.net/pictrs/image/02db5862-8341-45b0-a024-0bdea323db52.jpeg","banned":false,"published":"2024-01-30T11:44:48.964262Z","actor_id":"https://hexbear.net/u/Tomorrow_Farewell","local":false,"deleted":false,"bot_account":false,"instance_id":2562},"post":{"id":10328929,"name":"Mothematician post","url":"https://hexbear.net/pictrs/image/aac1969e-7f6d-458c-b437-33dd0cc6804b.jpeg","body":"This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.\n\nEnlighten him.","creator_id":6177520,"community_id":17845,"removed":false,"locked":false,"published":"2024-07-02T07:21:32.180198Z","deleted":false,"nsfw":false,"thumbnail_url":"https://hexbear.net/pictrs/image/6dcc9603-f729-4c65-abee-c66553fbc83a.jpeg","ap_id":"https://hexbear.net/post/2896015","local":false,"language_id":37,"featured_community":false,"featured_local":false},"community":{"id":17845,"name":"chapotraphouse","title":"chapotraphouse","description":"Banned? DM Wmill to appeal.\n\nNo anti-nautilism posts. See: [Eco-fascism Primer](https://hexbear.net/post/99029)\n\nSlop posts go in [c/slop](https://www.hexbear.net/c/slop). Don't post low-hanging fruit here.","removed":false,"published":"2021-01-01T01:01:43.573315Z","updated":"2024-12-01T18:50:21.595729Z","deleted":false,"nsfw":false,"actor_id":"https://hexbear.net/c/chapotraphouse","local":false,"icon":"https://hexbear.net/pictrs/image/14f93928-fc4a-45df-b220-49d54eedf397.jpeg","banner":"https://hexbear.net/pictrs/image/dc024c07-5aeb-4f93-b23f-e781b082f9cd.jpeg","hidden":false,"posting_restricted_to_mods":false,"instance_id":2562,"visibility":"Public"},"counts":{"comment_id":9605655,"score":5,"upvotes":5,"downvotes":0,"published":"2024-07-02T10:48:10.794677Z","child_count":5},"creator_banned_from_community":false,"banned_from_community":false,"creator_is_moderator":false,"creator_is_admin":false,"subscribed":"NotSubscribed","saved":false,"creator_blocked":false}]}