2. Prove each of the following statements using only the corresponding definition: (a) ) Let (an)o be a sequence of real numbers and let 0≤KEZ, LER. We define a sequence (bn)-o by: bn every 0≤n e Z. = an+k for Prove that the sequence (an)-o converges to L if and only if the sequence (bn)no converges to L. n=0 (b) Let (an)1 be a monotonically increasing sequence of real numbers. Prove that (an)n-1 converges in the extended sense, and that lim an = sup {an | neN}. n→∞ (Hint: follow two similar theorems that were proven in Calculus 1).
2. Prove each of the following statements using only the corresponding definition: (a) ) Let (an)o be a sequence of real numbers and let 0≤KEZ, LER. We define a sequence (bn)-o by: bn every 0≤n e Z. = an+k for Prove that the sequence (an)-o converges to L if and only if the sequence (bn)no converges to L. n=0 (b) Let (an)1 be a monotonically increasing sequence of real numbers. Prove that (an)n-1 converges in the extended sense, and that lim an = sup {an | neN}. n→∞ (Hint: follow two similar theorems that were proven in Calculus 1).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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