# Sequences and Uniform Convergence

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Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each nЄ I let

Fn(x) = ∫ x--> a fn(t)dt a<x<b

Show that {fn} infinity-->n-1 converges uniformly on [a,b]. (Hint: Use 9.2F)

Theorem 9.2F;

Let be a sequence of real-valued functions on a set E. Then is uniformly convergent on E ( to some function) if and only if given there exists such that

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Problem:

Let be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each let

Show that converges ...

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