Not to worry, I've managed to do it after all - I think it just boils down to proving the Uniform Convergence Theorem.
Problem
For let and be the subspace of continuous functions. Assume is a sequence with .
Prove that the limit is continuous.
Hint:
Use the -definition of continuity and the fact that is uniformly close to , for large enough , and that 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