{"id":1309,"date":"2017-10-26T12:23:25","date_gmt":"2017-10-26T16:23:25","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1309"},"modified":"2017-10-26T12:35:16","modified_gmt":"2017-10-26T16:35:16","slug":"muddiest-point-1024","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/26\/muddiest-point-1024\/","title":{"rendered":"Muddiest Point 10\/24"},"content":{"rendered":"

One of the things we did in the class was proving Theorem 4.4.7, the Uniform Continuity on Compact Sets. Theorem 4.4.7 states that a function is continuous on a compact set K is uniformly continuous on K.<\/p>\n

To me the prove in the class was hard to follow because the definition of compact set was not clear to me at that time. So I will review the concept of compact set and the prove again.<\/p>\n

A set \"K is compact if every sequence\u00a0\"(x_n) has a convergent subsequence converging in K. And another important theorem about compact set is Theorem 3.3.8 that states that the below statements are equivalent:<\/p>\n

    \n
  1. K is compact.<\/li>\n
  2. \u00a0K is closed and bounded.<\/li>\n
  3. Every open cover of K has a finite subcover.<\/li>\n<\/ol>\n

    The third statement is a little bit unfamiliar to us but the first two statements are very useful to us. And with that we can easily prove a continuous function f on a bounded subset of \"\mathbb{R}\" is uniformly continuous because a closed and bounded set is compact and then we can use Theorem 4.4.7 to prove that it is uniformly continuous.<\/p>\n

     <\/p>\n

    The prove of Theorem 4.4.7 proceeds with proving by contradiction. Since we want to prove that\u00a0\"f: is uniformly continuous, the contradiction to this is to prove that there exist two sequences\u00a0\"(x_n)\" and\u00a0\"\u00a0(y_n)\" in K such that lim \"|x_n yet\u00a0\"|f(x_n) for some\u00a0\"\epsilon_0.<\/p>\n

    Here we will utilize the concept of K being a compact set such that\u00a0the sequence \"(x_n)\" has a
    \nconvergent subsequence $latex(x_{n_k})$ with x = lim \"x_{n_k}\" also in K. Now we can use the Algebria Limit Theorem such that lim\"(y_{n_k}) lim\u00a0\"((y_{n_k}. So now we find a contradiction that lim\"(f(x_{n_k}) contradicting with\u00a0\"|f(x_n) for some\u00a0\"\epsilon_0. Thus we can prove that f is uniformly continuous on K.<\/p>\n

     <\/p>\n

    This prove utilizes different Theorem that we learned in the previous classes so it is helpful that we can review the old concepts that we learned.<\/p>\n","protected":false},"excerpt":{"rendered":"

    One of the things we did in the class was proving Theorem 4.4.7, the Uniform Continuity on Compact Sets. Theorem 4.4.7 states that a function is continuous on a compact set K is uniformly continuous on K. To me the prove in the class was hard to follow because the definition of compact set was […]<\/p>\n","protected":false},"author":3538,"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-1309","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-l7","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1309","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\/3538"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1309"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1309\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1309"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1309"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}