I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$.
I'm not familiar with Functional Analysis.
2026-04-03 07:33:51.1775201631
Is $C[0,1]$ locally Compact?
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All of these functions are in the ball of radius $1$ centered at $0$. They don't converge uniformly to any continuous function, nor does any subsequence converge uniformly to any continuous function (why?). Hence the ball is not precompact. Conclude from there.