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Math Help - Uniform Convergence

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    Uniform Convergence

    Let (f_n) be a sequence of functions that converges uniformly to f on A and that satisfies | f_n (x) | \le M for all n \in N and all x \in A.
    If g is continuous on the interval [-M,M], show that the sequence (g \circ f_n ) converges uniformly to g \circ f on A.

    Can anyone give me some hints to start the proof?
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    Quote Originally Posted by problem View Post
    Can anyone give me some hints to start the proof?
    Notice that g is uniformly continuous (as a continuous function on a bounded interval is uniformly continuous). Pick an epsilon, then |g(f_n(x))-g(f(x))|<\epsilon for some \delta such that |f_n(x)-f(x)|<\delta. Now what does uniform convergence of the f_n tell you about large enough n?
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