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(b) Prove that lim,→20,¤€A ƒ (x) exists and is equal to lim→0 Yk-

(b) Prove that lim,→20,¤€A ƒ (x) exists and is equal to lim→0 Yk-

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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Please solve 5(b)

#5 Let (fk)kEN be a sequence of functions from A C R™ to R" which converges uniformly on A lim,>xo,¤€A ƒk(x) exists for all k E N. Prove that (yk)kEN is a Cauchy sequence in R". (First prove that for all e > 0, to f : A → R". Let xo E A' and suppose that yk := (a) there exists N EN such that || fk(x) – fi(x)|| < e for all k, l > N and x E A. Then notice that ||Yk – Yı|| < ||Yk – fr(x)|| + || fr(x) – fi(x)||+ ||fi(x) – yı||-) (b) Prove that limg→x0,¤€A ƒ(x) exists and is equal to lim 00 Yk-
Transcribed Image Text:#5 Let (fk)kEN be a sequence of functions from A C R™ to R" which converges uniformly on A lim,>xo,¤€A ƒk(x) exists for all k E N. Prove that (yk)kEN is a Cauchy sequence in R". (First prove that for all e > 0, to f : A → R". Let xo E A' and suppose that yk := (a) there exists N EN such that || fk(x) – fi(x)|| < e for all k, l > N and x E A. Then notice that ||Yk – Yı|| < ||Yk – fr(x)|| + || fr(x) – fi(x)||+ ||fi(x) – yı||-) (b) Prove that limg→x0,¤€A ƒ(x) exists and is equal to lim 00 Yk-
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