Problem

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

Prove that the limit $\displaystyle f$ is continuous.

Hint:

Use the $\displaystyle (\epsilon ,\delta)$-definition of continuity and the fact that $\displaystyle f_n$ is uniformly close to $\displaystyle f$, for large enough $\displaystyle n$, and that $\displaystyle f_n$ 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