Prove that a sequence of continuous functions $f_n : [0,1] → \mathbb{R}$ converges uniformly to a continuous function f if and only if $f_n(x_n) → f(x)$ whenever $x_n → x$.
I have proved that limit n tends to infinity $f_n (x_n) =f (x)$ as $x_n$ tends to $x $ Am I right please Help to complete the solution
If $f_n$ does not converge uniformly to $f$, then there exists some $\epsilon > 0$, a subsequence $(n_k) \subset \mathbb{N}$ and a sequence $(x_{n_k})_k$ such that $|f_{n_k}(x_{n_k}) - f(x_{n_k})| > \epsilon$ for all $k$. Since $[0,1]$ is compact, we can take a further subsequence $(x_{n_{k_j}})_j$ of $(x_{n_k})_k$ that converges to some $x$. Then $|f_{n_{k_j}}(x_{n_{k_j}}) - f(x)| \ge |f_{n_{k_j}}(x_{n_{k_j}}) - f(x_{n_{k_j}})| - |f(x_{n_{k_j}}) - f(x)| > \epsilon/2$ for all large $j$, since the second term converges to zero by continuity of $f$.