# Math Help - Prove limit of sequence of functions is continuous

1. ## 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.

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Um, I have no idea how to go about this - despite the hint. Any help people can give would be great

2. Not to worry, I've managed to do it after all - I think it just boils down to proving the Uniform Convergence Theorem.