Foundations of Analysis
Foundations of Analysis MATH 4200
Utah State University
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This 1 page Class Notes was uploaded by Benton Yundt on Wednesday October 28, 2015. The Class Notes belongs to MATH 4200 at Utah State University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/230423/math-4200-utah-state-university in Mathematics (M) at Utah State University.
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Date Created: 10/28/15
Math 4200 Theorem Let anbe a sequence of real numbers Then lim an a if and only if given any open interval r s containing a r s contains all but nitely many terms of the sequence an Proof 1 Suppose lim an a Let r s be an open interval containing a Let 8mina r 3 6 There exists M gt0 suchthat if ngtM then If ngtMthen a 8ltan lta8 IfngtMthen a a rltan ltas a an alt8 If n gt M then r lt an lt s Therefore r s contains all but nitely many terms of the sequence 11 Suppose that given any open interval r 3 containing a r 3 contains all but nitely many terms of the sequence an Let 8 gt 0 Consider the interval a 8 a 8 If all but nitely many terms of the sequence are contained in this interval then there eXists M gt 0 such that if n gtM then a 8 lt an lt a 8 So there eXists M gt 0 such that if n gtM then an alt8
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