The problem is as follows.
Let {$f_n$} is defined on $[0,1]$ and suppose that $lim_{n-> infinity}$ $f_n(x_n)=0$ for convergent sequence in $[0,1]$, {$x_n$}.
Show that {$f_n$} is converges uniformly to $f=0$ on $[0,1]$.
I think using the bolzano-weierstrass theorem but i get stuck here. I'd like to show that Every sequence {$x_n$} in $[0,1]$, $lim_{n->infinity} |f_n(x_n)|=0$ . Please give me full explanation.
If $f_n$ does not converges to $0$ uniformly the there exist $\epsilon >0$, integers $n_1<n_2...$ and point $x_k$ such that $|f_{n_k}(x_k)| >\epsilon$ for all $k$. There is a subsequence, say $y_k$, of $(x_k)$ which converges to some point $x$. But then the hypothesis is violated since $f_n(y_n)$ does not tend to $0$.