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Suppose (fk)kɛN is a sequence of functions from A C R™ to R" converging uniformly on A to f : A → R". Prove that if each fk is uniformly continuous on A, then so is f. (Notice that || f(x) – f(y)|| < ||F(x) – fx(x)|| + || fr(x) – fr(y)|| + || fx(y) – f(y)||-)

Suppose (fk)kɛN is a sequence of functions from A C R™ to R" converging uniformly on A to f : A → R". Prove that if each fk is uniformly continuous on A, then so is f. (Notice that || f(x) – f(y)|| < ||F(x) – fx(x)|| + || fr(x) – fr(y)|| + || fx(y) – f(y)||-)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Suppose (fk)kEN is a sequence of functions from A C R™ to R" converging uniformly on A to f : A → R". Prove that if each fk is uniformly continuous on A, then so is f. (Notice that || f(x) – f(y)|| < || F(x) – fr(x)|| + || fr(x) – fk (y)|| + || fx (y) – f(y)||-) #3
Transcribed Image Text:Suppose (fk)kEN is a sequence of functions from A C R™ to R" converging uniformly on A to f : A → R". Prove that if each fk is uniformly continuous on A, then so is f. (Notice that || f(x) – f(y)|| < || F(x) – fr(x)|| + || fr(x) – fk (y)|| + || fx (y) – f(y)||-) #3
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