Hi, I've been working on:

Construct a bounded sequence of continuous functions $\displaystyle f_n:[0,1] \rightarrow \mathbb{R} \\ s.t. \left| \left| f_n - f_m \right| \right| = sup \left| f_n(x) - f_m(x) \right| = 1, n \neq m, x \in [0,1] $

Can such a sequence be equicontinuous?

So far I have the example that $\displaystyle f_n(x) = sin(nx) $ but I don't know how to handle the equicontinuity. I know this particular family of functions does not have a uniformly convergent subsequence so it can't be equicontinuous, but I don't know how to rigorously prove it. Any help would be much appreciated!