Finding a bounded sequence of $C[0,\pi]$ such that it has no cauchy subsequnce

Question

Consider $(X,\|\cdot\|) = (C[0,\pi],\|\cdot\|_{2})$ where $\|f\|_{2} = \left (\int_{0}^{\pi} |f(t)|^{2} \,\mathrm{d}t)\right )^{\frac{1}{2}}$ How can I find a bounded sequence $(f_{n}) \in X$ which has no Cauchy subsequence? I'm aware of the question (Sequence in $C[0,1]$ with no convergent subsequence) but their example doesn't work for this norm.

Answer 1

Consider for instance $f_n(x)=\sin(n\,x)$.

Answer 2

If, in a normed space, every bounded sequence has a Cauchy subsequence then the unit ball is precompact which implies that the space is finite dimensional.