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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Transcribed Image Text: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)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 | n E N}.
n→∞
(Hint: follow two similar theorems that were proven in Calculus 1).
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