In this post, I will discuss the following definitions we covered in Thursday’s (9\/7) class:<\/p>\n
Chapter 1: Theorem 1.5.6<\/strong><\/p>\n Chapter 2: Sequence, Convergence of a Sequence, Theorem 2.2.7 (Uniqueness of Limits) and Bounded<\/strong><\/p>\n <\/p>\n Theorem 1.5.6<\/strong> (i) Q\u00a0\u2208 N (Q is countable) (ii) R is uncountable<\/p>\n (Proof Template)<\/em><\/p>\n (i) Set A1 = {0} and for each n\u00a0\u2265 2, let An be the set given by An = {\u00b1 p\/q: where where p, q \u2208 N are in lowest terms with p + q = n}<\/p>\n The first few of these sets look like A1 = {0}, A2 ={1\/1, -1\/1}, A3= {1\/2, -1\/2, 2\/1, -2\/1} and A4 ={1\/3, -1\/3, 3\/1, -3\/1}<\/p>\n We can observe that each An is finite and every rational number appears in exactly one of these sets. Our 1\u20131 correspondence with N is then achieved by consecutively listing the elements in each An.<\/p>\n (ii) Proof by contradiction. Assume that there does exist a 1\u20131, onto function f : N \u2192 R. Let x1 = f(1), x2 = f(2)… R = {x1, x2, x3, x4, . . .} We can use the Nested Interval Property (Theorem 1.4.1) to produce a real number that is not there.<\/p>\n <\/p>\n Sequence:<\/strong> A sequence is a function whose domain is N<\/p>\n <\/p>\n Convergence of a Sequence:<\/strong>\u00a0A sequence (a_n) converges to a real number a if, for every positive number \u03b5, there exists an N \u2208 N such that whenever n \u2265 N it follows that |a_n \u2212 a| < \u03b5.<\/p>\n (i) lim a_n = a or lim_(n\u2192\u221e) a_n = a<\/p>\n (ii) (a_n) converges in R<\/p>\n Notation: \u2200\u00a0\u03b5 > 0,\u00a0\u2203n_0 \u2208 N such that |a_n \u2212 a| <\u00a0\u03b5 for all n\u00a0\u2265 n_0.<\/p>\n <\/p>\n Convergence of a Sequence (Topological Version):\u00a0<\/strong>\u2200\u00a0\u03b5 > 0,\u00a0\u2203 n_0 \u2208 N such that a_n \u2208 V_\u03b5(a) for all n\u00a0\u2265 n_0.<\/p>\n <\/p>\n Theorem 2.2.7 (Uniqueness of Limits):\u00a0<\/strong>The limit of a sequence, when it exists, must be unique.<\/p>\n Notation: (a_n)\u00a0\u2192 a and (a_n)\u00a0\u2192 b, then a = b.<\/p>\n (Proof Template)<\/em><\/p>\n Recalls Theorem 1.2.6\u00a0\u2200 a, b \u2208 R, a = b\u00a0\u2194\u00a0\u00a0<\/strong>\u2200\u00a0\u03b5 > 0, |a \u2212 b| <\u00a0\u03b5.<\/p>\n PF:<\/span><\/strong>\u00a0let\u00a0\u03b5 > 0. Since\u00a0(a_n)\u00a0\u2192 a, \u2203 N1 \u2208 N such that |a_n \u2212 a| <\u00a0\u03b5\/2 for n\u00a0\u2265 N1. Since\u00a0(a_n)\u00a0\u2192 b, \u2203 N2 \u2208 N such that |a_n \u2212 b| <\u00a0\u03b5\/2\u00a0for n\u00a0\u2265 N2. Pick n_0 = max{N1, N2} and consider\u00a0n\u00a0\u2265 n_0. |a \u2212 b| = |a – a_n + a_n \u2212 b| \u2264 |a – a_n| + |a_n – b| <\u00a0\u03b5\/2. Since n \u2265 n_0 \u2265 N1,\u00a0|a – a_n| <\u00a0\u03b5\/2.\u00a0Since n \u2265 n_0 \u2265 N,\u00a0|a_n – b| <\u00a0\u03b5\/2. Thus, since\u00a0\u03b5 > 0 is arbitrary, we conclude a = b.<\/p>\n <\/p>\n Bounded: <\/strong>(a_n) is bounded if\u00a0\u2203 M > 0 so that |a_n|\u00a0\u2264 M,\u00a0\u2200 n \u2208 N.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" In this post, I will discuss the following definitions we covered in Thursday’s (9\/7) class: Chapter 1: Theorem 1.5.6 Chapter 2: Sequence, Convergence of a Sequence, Theorem 2.2.7 (Uniqueness of Limits) and Bounded Theorem 1.5.6 (i) Q\u00a0\u2208 N (Q is countable) (ii) R is uncountable (Proof Template) (i) Set A1 = {0} and for […]<\/p>\n","protected":false},"author":3536,"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-777","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-cx","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/777","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\/3536"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=777"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/777\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}