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Math Help - Prove limit of sequence of functions is continuous

  1. #1
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    Prove limit of sequence of functions is continuous

    Problem

    For f,g\in B([a,b]) let d(f,g):=sup_{x\in [a,b]}|f(x)-g(x)| and C([a,b])\subset B([a,b]) be the subspace of continuous functions. Assume f_n\in C([a,b]) is a sequence with f_n\rightarrow f\in B([a,b]).
    Prove that the limit f is continuous.

    Hint:
    Use the (\epsilon ,\delta)-definition of continuity and the fact that f_n is uniformly close to f, for large enough n, and that f_n is continuous. Combine all this to create a precise proof.

    --------------------------

    Um, I have no idea how to go about this - despite the hint. Any help people can give would be great
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  2. #2
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    Not to worry, I've managed to do it after all - I think it just boils down to proving the Uniform Convergence Theorem.
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