{"id":543,"date":"2013-09-28T18:48:47","date_gmt":"2013-09-28T22:48:47","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/groisser\/?page_id=543"},"modified":"2026-03-19T08:13:06","modified_gmt":"2026-03-19T12:13:06","slug":"hw4","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/groisser\/classes\/4211_f13\/hw4\/","title":{"rendered":"MAA 4211 Assignment 4"},"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 4<\/h1>\n\n\n\n\n\n\n\n\n\n<p>MAA 4211 Assignment 4<br>\nDue date: Monday, 10\/7\/13\u00a0<\/p>\n\n\n\n<p><i>Last update made by D. Groisser Fri Oct 4 15:25:28 EDT 2013 <\/i><\/p>\n\n\n\n\n\n<p>\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 below 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\/ 1bc, 3, 4, 6, 7. <strong> Of these, hand in only 1c, 3, 4, 6.<\/strong> Note:\n<ul>\n<li>In 1b: a sequence {<i>x<sub>n<\/sub><\/i>} of real numbers is <i>bounded<\/i> if there exists <i>M<\/i>\u2208<strong>R<\/strong> such that |<i>x<sub>n<\/sub><\/i>| \u2264 <i>M<\/i> for all <i>n<\/i>\u2208<strong>N<\/strong>.<\/li>\n<li>In problems like 4 and 6, keep in mind that &#8220;proof by picture&#8221; is not a valid method of proof. In these two problems, you will need to show <i>algebraically<\/i> that open balls of certain centers and radii (which you have to figure out) are contained in certain sets.<\/li>\n<li>In Rosenlicht, <i>E<sup>n<\/sup><\/i> means Euclidean <i>n<\/i> space: the metric space (<strong>R<\/strong><sup><i>n<\/i><\/sup>, <i>d<\/i>), where <i>d<\/i> is the Euclidean metric (the square root of the sum of the squares of differences of coordinates). (See the last paragraph of p. 34.) So in problems 4 and 6, <i>E<\/i><sup>2<\/sup> is the usual <i>xy<\/i> plane (or <i>x<\/i><sub>1<\/sub><i>x<\/i><sub>2<\/sub> plane) with the distance-formula that you&#8217;re used to. In these problems, you may use the notation (<i>x<\/i>, <i>y<\/i>) instead of (<i>x<\/i><sub>1<\/sub>,<i>x<\/i><sub>2<\/sub>), but state that you&#8217;re doing this, so that I know what you mean from the start.<\/li>\n<\/ul>\n<\/li><li><strong>B1:<\/strong> <strong>(Hand this one in.)<\/strong> Define a metric <i>d<\/i> on the set of rational numbers <strong>Q<\/strong> by <i>d<\/i>(<i>x,y<\/i>) = |<i>x \u2013 y<\/i>| (the restriction to <strong>Q<\/strong> of the standard metric on <strong>R<\/strong>). Give an example, with proof, of a nonempty, proper subset of (<strong>Q<\/strong>,<i>d<\/i>) that is both open and closed <i>in this metric space<\/i>. (Do not expect your subset to be either open or closed in <strong>R<\/strong>, let alone both open <i>and<\/i> closed. There is no nonempty, proper subset of <strong>R<\/strong> that is both open and closed with respect to the standard metric.)<\/li>\n<li><strong>C:<\/strong> Read the handout &#8220;Interiors, Closures, and Boundaries&#8221; posted on the <a href=\"https:\/\/people.clas.ufl.edu\/groisser\/classes\/4211_f13\/handouts\/\"> Miscellaneous Handouts<\/a> page. (You may ignore fact #13 until we&#8217;ve defined <i>convergent sequences<\/i>.) Prove facts (4)-(5), (7)-(10), and (14)-(21) stated in the handout. <strong> Of these, hand in the proofs of only (9), (10), (16), (20), and (21). <\/strong> When working on any of these, you may assume any of the facts listed <i>earlier<\/i> in the handout, but not those listed later. On Friday Oct. 4, I&#8217;ll prove facts (11) and (12), and possibly some of the others from (7)-(21), but don&#8217;t wait to see which ones I do in class to try doing these on your own.<\/li><\/ul>\n\n\n\n\n\n\n\n<p><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":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-543","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/543","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=543"}],"version-history":[{"count":10,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/543\/revisions"}],"predecessor-version":[{"id":5247,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/543\/revisions\/5247"}],"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=543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}