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# Class Note for MATH 323 with Professor Laetsch at UA 2

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 42 views.

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Date Created: 02/06/15
Math 3231 Lesson 53a 11 Equivalent De nitions of Limit Prove that the following two de nitions of convergence are equivalent to each other and also to the original de nition in the textbook B00kDefiniti0nV8gt0 EINONaVnONngtNgtlsn Lllt8 Definition 1 Sn converges to L means t0hat for every real number 8 gt 0 we can nd a positive real number N such that N 8 for every natural number 11 EN l Sn Ll lt 8 Definition 2 Sn converges to L means that for every real number k gt 0 we can nd a natural number N such that N k for every natural number n gt N l Sn L l lt k Proof of equivalence Given a sequence Sn that has a limit L Book defn implies defn 1 Consider an s gt 0 By the book de nition there exists a real number N such that for S 14 lt 5 Choose M N 1 Then consider a natural number 112 M Since 112 M N l gt N l 3 Ll lt 8 from the book de nition every natural number n gt N Defn 1 implies defn 2 Suppose k is a real number greater than 0 De nition 1 implies that there exists a real number N such that for all natural numbers n 2 N s 14 lt k By the Archimedean property of the natural numbers for every real number there exists a natural number that is greater than that real number Choose M to be a natural number greater than N Consider an arbitrary natural number n gt M Since 11 gt M gt N implies n gt N lsquot Ll ltk by de nition 1 Defn 2 implies book defn Consider an s gt 0 By de nition 2 there exists a natural number N such that for all natural numbers n gt N Is Ll lt 8 Since N is also a real number choose M N Consider an arbitrary natural number n gtM By the de nition 2 this implies l Sn Ll lt 8

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