Suppose we have a sequence of functions defined on $S$, $f_n$ that converges uniformly to $f$. Then if $x_n$ is any sequence in the interval can we say that $f_n(x_n)-f(x_n)$ goes to zero?
By definition of uniform convergence, given any positive $\epsilon$ we have, $|f_n-f|<\epsilon$ for every point in $S$ for some $n>N$
Now let $x_n$ be a sequence in interval. In particular take $\epsilon_n=1/n$. There exist $N$ such that $|f_n(x_n)-f(x_n)|<\epsilon_n$ for all $n>N$
But I am not able to proceed from here?
Thanks for help.