{"id":1660,"date":"2013-12-01T21:51:57","date_gmt":"2013-12-02T02:51:57","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/groisser\/?page_id=1660"},"modified":"2026-03-19T08:13:08","modified_gmt":"2026-03-19T12:13:08","slug":"hw5","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/groisser\/classes\/4211_f13\/hw5\/","title":{"rendered":"MAA 4211 Assignment 5"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h1 class=\"wp-block-heading\">MAA 4211 Assignment 5<\/h1>\n\n\n\n\n\n\n\n\n\n<p class=\"wp-block-paragraph\">MAA 4211 Assignment 5<br>\nDue date: Friday 10\/25\/13.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><i>Last update made by D. Groisser Tue Oct 22 00:54:22 EDT 2013 <\/i><\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">\u00a0<br>\nYou are required to do <strong> all <\/strong> of the problems below. You will not be required to hand them all in. I&#8217;ve indicated which ones you do have to hand in. Don&#8217;t make the mistake of thinking that I&#8217;m collecting only the problems I think are important.The &#8220;due date&#8221; above is the date that your written-up problems should be handed in, but don&#8217;t wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.<\/p>\n\n\n\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li><strong>A:<\/strong> Rosenlicht pp. 61\u201363\/ 8,12,13,14,23. <strong> Of these, hand in only 8, 13, and 23<\/strong>. For #23, hand in only the proofs for &#8220;<i>a<sub>n<\/sub> + b<sub>n<\/sub><\/i>&#8221; and &#8220;<i>c<sub>n<\/sub> a<sub>n<\/sub><\/i>&#8220;; don&#8217;t hand in the proof for &#8220;<i>a<sub>n<\/sub> &#8211; b<sub>n<\/sub><\/i>&#8220;. Prior to doing #13, do B3 in the non-book problems. You may assume the result of B3 to do #13, whether or not you are successful with B3.Definition for #8: A <i>reordering<\/i> of a sequence <i>{p<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i> is a sequence <i>{q<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i> such that for some bijection <i>f<\/i>: <strong>N<\/strong>\u2192 <strong>N<\/strong> we have <i>q<sub>n<\/sub> = p<sub>f(n)<\/sub><\/i> for all <i>n<\/i>\u2208<strong>N<\/strong>. (In &#8220;<i>{p<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i>&#8220;, the superscript &#8220;\u221e&#8221; is meant to be sitting over the subscript &#8220;<i>n=1<\/i>&#8220;, but I don&#8217;t know how to achieve this in HTML.) Problem 8 is asking you to prove that if <i>{q<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i> is a reordering of a convergent sequence <i>{p<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i> in a metric space, then <i>{q<sub>n<\/sub>}<sub>{n=1}<\/sub><sup>\u221e<\/sup><\/i> converges and lim<sub><i>n<\/i> \u2192\u221e<\/sub> <i>q<sub>n<\/sub><\/i> = lim<sub><i>n<\/i> \u2192\u221e<\/sub> <i>p<sub>n<\/sub><\/i>.<\/li>\n<li><strong>B:<\/strong> Click here for <a href=\"https:\/\/people.clas.ufl.edu\/groisser\/content-removed\/\">non-book problems<\/a>. <strong> Of these, hand in only B1ab, B2, B3, and B5b<\/strong>.<\/li><\/ul>\n\n\n\n\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"..\/\"> Back to class home page <\/a><\/p>\n\n\n\n\n\n\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":245,"featured_media":0,"parent":241,"menu_order":11,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-1660","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1660","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/users\/245"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/comments?post=1660"}],"version-history":[{"count":8,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1660\/revisions"}],"predecessor-version":[{"id":5310,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1660\/revisions\/5310"}],"up":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/241"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/media?parent=1660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}