{"id":1136,"date":"2017-10-19T02:19:20","date_gmt":"2017-10-19T06:19:20","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1136"},"modified":"2017-10-19T02:19:20","modified_gmt":"2017-10-19T06:19:20","slug":"muddiest-point-101217","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/19\/muddiest-point-101217\/","title":{"rendered":"Muddiest Point 10\/12\/17"},"content":{"rendered":"

One of the things we did in this class was introduce and go through the proof for Theorem 4.2.3 (Sequential Criterion for Functional Limits). The theorem states: Given a function \"f:A and \"c, the following are equivalent:<\/p>\n

i.) \"lim_{x<\/p>\n

ii.)\"\forall\" sequences \"(x_n) such that \"x_n and \"(x_n) it follows that \"(f(x_n)) For me, I think that the muddiest point lies in fully understanding the logic in the proof.<\/p>\n

For me, I think that the muddiest point lies in fully understanding the logic in the proof, so I’m going to go through the intuition needed to understand the proof. For the \"\Rightarrow\" direction of the proof, we assume part i and prove part ii by using the topological definition of a function limit. We start with \"\varepsilon, and then say that there exists a \"V_\delta that contains all x’s that are not equivalent to c. From the definition, it follows that \"f(x) At this point, it is important to distinguish what we still need to show. We want to show that\u00a0\"f(x_n) so we can see that in order to bridge the gap between\u00a0what we have and what we want, we need N for which the statement is true with \"n so that\u00a0\"f(x_n) is true.<\/p>\n

The other direction of the proof is a proof by contraposition. We assume that ii is true and negate i, so it says $latex\u00a0lim_{x \\to c} f(x) \\neq L.$ Now, we can use the topological definition of a functional limit to help us here. From the negation of ii, we know that there exists a \"\varepsilon_o such that \"\forall \"\exists such that \"f(x) Since we want to contradict our negation, we proceed by picking a specific \"\delta\" and a point \"x_n\". These are used to show that \"f(x_n) However, this means that \"x_n so \"f(x_n)\" does not converge to \"L.\" This contradicts ii, which was assumed to be true, so we see that i holds to be true.<\/p>\n

Overall, the proof of Theorem 4.2.3 is easier to complete when the topological definition of a function limit is fully understood. Also, I found that the graph on the top of page 116 in the book especially useful when trying to visualize what is happening with the proof.<\/p>\n","protected":false},"excerpt":{"rendered":"

One of the things we did in this class was introduce and go through the proof for Theorem 4.2.3 (Sequential Criterion for Functional Limits). The theorem states: Given a function and , the following are equivalent: i.) ii.) sequences such that and it follows that For me, I think that the muddiest point lies in […]<\/p>\n","protected":false},"author":2206,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58822],"tags":[],"class_list":["post-1136","post","type-post","status-publish","format-standard","hentry","category-muddiest-point"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-ik","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/2206"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1136"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1136\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}