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Advanced Engineering Mathematics
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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(24) Let U be a non-principal ultrafilter over w. Suppose (xn : n <w) is a sequence of real numbers. We say that r is the U-limit of (xn : n < w) and write U lim In = x iff for every & > 0, {n <w: |xn – x| < ɛ} €U. Show the following. (a) If lim xn = x, then U lim xn = x. (b) If U lim xn = a and U lim xn = b, then a = b. n (c) If (xn : n <w) is bounded in R, then there exists a unique real x such that U lim xn = x.
Transcribed Image Text:(24) Let U be a non-principal ultrafilter over w. Suppose (xn : n <w) is a sequence of real numbers. We say that r is the U-limit of (xn : n < w) and write U lim In = x iff for every & > 0, {n <w: |xn – x| < ɛ} €U. Show the following. (a) If lim xn = x, then U lim xn = x. (b) If U lim xn = a and U lim xn = b, then a = b. n (c) If (xn : n <w) is bounded in R, then there exists a unique real x such that U lim xn = x.
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