Foundations of Analysis
Foundations of Analysis MATH 4200
Utah State University
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This 2 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 E. Heal in Fall. Since its upload, it has received 28 views. For similar materials see /class/230412/math-4200-utah-state-university in Mathematics (M) at Utah State University.
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Date Created: 10/28/15
Math 4200 De nition A sequence of real numbers is afunction f J gt R De nition A sequence a7 is said to be bounded provided the set a n E J isbounded De nition A sequence a7 is said to be bounded above below provided the set a7 n E J is bounded above below De nition Let an be a sequence We say aquot has a limit if and only if there exists a real number L such that for every 6 gt 0 there exists M gt 0 such that for any n in J ifngtMthen an L lte The number L is called the limit of the sequence a7 Notation If an is a sequence with the number L as a limit we write lim an L and say the limit of a as n tends to in nity is L H00 1 Establish the existence or nonexistence of limits for the following sequences TL n21 b 1 71n 9 v 2Given e 00000057 find a suitable M such that for all positive integers n gt M 7 2 lt6 3Prove that hm an L if and onlyif hm b 0 where for eachnin J bn an 7 L 4 If aw and bn are two convergent sequences satisfying an 3 b for all n can you conclude that m an limbn 7 5 Let a and bn be sequences having limits A and B respectively Prove that a b converges to A B 6 Let a and bn be sequences having limits A and B respectively Prove that a b converges to A B 7 Suppose an converges to 0 and b is bounded Show that aw bn converges to 0