{"id":1466,"date":"2017-11-14T23:12:08","date_gmt":"2017-11-15T04:12:08","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1466"},"modified":"2017-11-14T23:12:08","modified_gmt":"2017-11-15T04:12:08","slug":"muddiest-point-1114","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/14\/muddiest-point-1114\/","title":{"rendered":"Muddiest Point 11\/14"},"content":{"rendered":"

For me, the muddiest point of this lecture was understanding the order of choosing \"N\" and \"\delta\". I will illustrate this in the proof of Theorem 6.2.6.<\/p>\n

Statement of Proof: If \"f_n uniformly on \"A\" and \"f_n\" continuous on \"A\" for all \"n\", then\u00a0\"f\" is continuous on\u00a0\"A\" as well.<\/p>\n

Proof: Let\u00a0\"\epsilon. Choose\u00a0\"N\" such that\u00a0\"||f_N-f||_{\infty,A}. Choose\u00a0\"\delta\" such that\u00a0\"|x-c| implies\u00a0\"|f_N(x)-f_N(c)|. By the triangle inequality,\u00a0\"|f(x)-f(c)|.<\/p>\n

It is necessary to choose\u00a0\"N\" before\u00a0\"\delta\" because\u00a0\"\delta\" relies on\u00a0\"N\". Imagine trying to choose\u00a0\"\delta\" first. You wouldn’t be able to choose a\u00a0\"\delta\" that would help solve the problem because the correct \"\delta\" relies on\u00a0\"N\".<\/p>\n","protected":false},"excerpt":{"rendered":"

For me, the muddiest point of this lecture was understanding the order of choosing and . I will illustrate this in the proof of Theorem 6.2.6. Statement of Proof: If uniformly on and continuous on for all , then\u00a0 is continuous on\u00a0 as well. Proof: Let\u00a0. Choose\u00a0 such that\u00a0. Choose\u00a0 such that\u00a0 implies\u00a0. By the […]<\/p>\n","protected":false},"author":3530,"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":[58820,58822],"tags":[],"class_list":["post-1466","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-muddiest-point"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-nE","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1466","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\/3530"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1466"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1466\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1466"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1466"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1466"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}