Let me consider $C[0,1]$ with the metric $d(f,g)=\max\{|f(t)-g(t)|: t\in [0,1]\}$ and define $D=\{f\in C([0,1]): d(f,0)\leq 1\}$ I want to show that this is not compact.
I know that it is enough to show that it is not sequentially compact, therefore I thought that if I find a sequence in $D$ s.t. there exists no convergent subsequence I would be done right?
But if I take $f_n(x)=x^n$ for all $x\in [0,1]$ then clearly $f_n\in D$. But as $n\rightarrow \infty$ we know that $f_n\rightarrow 0$ if $x\in [0,1)$ and $f_n\rightarrow 1$ if $x=1$. But then the limit is discontinuous. But this would then show that there exists no convergent subsequence with limit in $D$
Is this correct?