This question is taken from Saxe K -Beginning Functional Analysis.
Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.
(It's assumed that we are using the uniform norm).
I've been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)>c$. Should be a nice example of this, please help me!
Best Answer
Consider $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$ (closed unit ball), but no subsequence of $\{f_n\}$ converges in $C[0,1]$ (with the sup norm).