{"id":852,"date":"2017-09-21T01:04:14","date_gmt":"2017-09-21T05:04:14","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=852"},"modified":"2017-09-21T01:04:14","modified_gmt":"2017-09-21T05:04:14","slug":"what-happened-919","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/09\/21\/what-happened-919\/","title":{"rendered":"What Happened 9\/19"},"content":{"rendered":"

Last Thursday we ended class with beginning to prove Theorem 2.6.4 which states that a sequence \"(a_n)\" converges \"\iff is Cauchy. We picked up on Tuesday with a bit of review of this proof: We are assuming \"(a_n)\" is Cauchy and wanting to prove that \"(a_n)\" converges. Using Lemma 2.6.3 and Bolzano – Weierstress we concluded that there exists a convergent sub-sequence \"(a_{n_k}\" such that \"(a_{n_k}. From here Dr.LeCrone brought us through his thought process and ultimately showed the class how to piece together the remainder of the proof. He did so in the following steps:<\/p>\n

1. What is our claim? \"(a_n)<\/p>\n

2. What are the knowns? \"(a_n)\" is Cauchy, \"(a_{n_k}<\/p>\n

3. What do we WTS? \"\forall such that \"|a_n<\/p>\n

4. What will the outline of the proof look like?<\/p>\n

– Let \"\epsilon …<\/p>\n

– Choose N as already given by Cauchy condition<\/p>\n

– Let \"n, thus we have $|a_n – a| < \\epsilon$<\/p>\n

5. Now we need to do some of our side work and fill in the blanks between this proof. I encourage you to go over the class notes to see exactly how Dr.LeCrone did this. We basically used the triangle inequality to expand what we knew and from there fixed a \"K where we got these values from our expansion using the inequality.<\/p>\n

 <\/p>\n

Overall, he walked us through building this important proof and showed us the consequences this proof gave:<\/p>\n

1. \"(a_n)\" diverges \"\iff is not Cauchy<\/p>\n

2. The negation of Cauchy is a much nicer way of proving divergence<\/p>\n

 <\/p>\n

We are now done with sequences and now onto series!<\/p>\n

The remainder of class was spent going over Theroems 2.7.2 and 2.7.3 which again I encourage you to look over.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Last Thursday we ended class with beginning to prove Theorem 2.6.4 which states that a sequence converges is Cauchy. We picked up on Tuesday with a bit of review of this proof: We are assuming is Cauchy and wanting to prove that converges. Using Lemma 2.6.3 and Bolzano – Weierstress we concluded that there exists […]<\/p>\n","protected":false},"author":3525,"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":[58821],"tags":[],"class_list":["post-852","post","type-post","status-publish","format-standard","hentry","category-what-happened-today"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-dK","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/852","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\/3525"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=852"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/852\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}