{"id":723,"date":"2017-09-05T02:10:22","date_gmt":"2017-09-05T06:10:22","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=723"},"modified":"2017-09-05T02:10:22","modified_gmt":"2017-09-05T06:10:22","slug":"1st-class-tuesday-29th-september","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/09\/05\/1st-class-tuesday-29th-september\/","title":{"rendered":"1st Class &#8211; Tuesday, 29th September"},"content":{"rendered":"<p>In this post, I will discuss the following definitions we covered in the class: <em><strong>Upper Bound<\/strong>, <strong>Least Upper Bound<\/strong> (Supremum), and <strong>Axiom of Completeness<\/strong> (AoC).<\/em>\u00a0I will try to provide a proof template for most of the definitions I state.<\/p>\n<p><em><strong>1. Upper Bound<br \/>\n<\/strong><\/em>If set\u00a0<em>A<\/em>\u00a0is a subset of <strong>\u211d<\/strong>, then a number <em>b<\/em> is called an upper bound for <em>A\u00a0<\/em>if it is equal to or greater than all the elements of <em>A<\/em>. We say that <em>A<\/em>\u00a0is bounded above if such a number exists.<\/p>\n<p><span style=\"text-decoration: underline\">Proof template:<br \/>\n<\/span> Let <em>a<\/em> in A and <em>b<\/em> in <strong>\u211d<\/strong>, then if<em> b<\/em> is greater than or equal to <em>a <\/em>we say that<em>\u00a0b<\/em> is an upper bound for <em>A<\/em>. This holds because <em>a<\/em> was chosen arbitrarily.<\/p>\n<p><em><strong>2. Least Upper Bound<br \/>\n<\/strong><\/em>A real number <em>s<\/em> is the least upper bound for a set <em>A<\/em> in<strong> \u211d<\/strong> if it meets the following criteria:<br \/>\n(i) <em>s <\/em>is an upper bound for <em>A,<\/em><br \/>\n(ii) if <em>b<\/em> is any upper bound for <em>A<\/em>, then <em>s<\/em> is less than or equal to <em>b<\/em>.<\/p>\n<p><span style=\"text-decoration: underline\">Proof Template:<\/span><br \/>\nLet <em>a<\/em> in <em>A<\/em>, then if <em>s<\/em> is greater than or equal to all the elements in <em>A<\/em> it is an upper bound for <em>A<\/em>. Thus, <em>s <\/em>meets the first criteria of the above stated definition. Now let\u2019s assume there exist any other upper bound <em>b<\/em>. If <em>s<\/em> is less than or equal to <em>b,<\/em> then we can conclude that <em>s<\/em> is the least upper bound for <em>A<\/em>.<\/p>\n<p><em><strong>3. Axiom of Completeness<br \/>\n<\/strong><\/em>Every nonempty set of real numbers that is bounded above has a least upper bound.<\/p>\n<p>&nbsp;<\/p>\n<p><em>Sami<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this post, I will discuss the following definitions we covered in the class: Upper Bound, Least Upper Bound (Supremum), and Axiom of Completeness (AoC).\u00a0I will try to provide a proof template for most of the definitions I state. 1. Upper Bound If set\u00a0A\u00a0is a subset of \u211d, then a number b is called an [&hellip;]<\/p>\n","protected":false},"author":3533,"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":[58823],"tags":[],"class_list":["post-723","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-bF","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/723","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\/3533"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=723"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/723\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=723"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}