2. Prove each of the following statements using only the corresponding definition: (a) n=0 :) Let (an) 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)-o converges to L. (b) Let (an) be a monotonically increasing sequence of real numbers. Prove that (an)n-1 converges in the extended sense, and that lim an = sup {an | ne N}. 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) n=0 :) Let (an) 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)-o converges to L. (b) Let (an) be a monotonically increasing sequence of real numbers. Prove that (an)n-1 converges in the extended sense, and that lim an = sup {an | ne N}. n→∞ (Hint: follow two similar theorems that were proven in Calculus 1).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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