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Question: exercise 625 using the cauchy criterion for convergent sequences of...

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Exercise 6.2.5. Using the Cauchy Criterion for convergent sequences of real numbers (Theorem 2.6.4), supply a proof for Theorem 6.2.5. (First, define a candidate for f(x), and then argue that fn → f uniformly.)
Theorem 6.2.5 (Cauchy Criterion for Uniform Convergence). A se- quence of functions (fn) defined on a set A C R converges uniformly on A if and only if for every e>0 there erists an N E N such that fn(x)-fm(x)<e whenever m, n 2 N and z E A
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