Suppose $K$ is a compact set in $\mathbb{R}$, $f_n$ is a sequence of functions such that $f_n$ converges pointwise to a continuous function $f$, does it imply $f_n$ converges uniformly to $f$ in $K$ ?
Edit: (addition of Hypothesis)$f_n$ is continuous for each $n$
If it is not true, what example justifies it ?
No, if there is no condition that $f_n$ are continuous. Counterexample: $K = [0,1]$, $f_n(x)$ $=$ $0$ for $x\neq1/n$, $1$ for $x = 1/n$, $f(x) \equiv 0$.
EDIT: For continuous functions it is still not true. $K$ and $f$ as above, and $f_n(x) = 0$ for x < $1/2n$, $0$ for $x > 3/2n$, $1$ for $1/n$ and linear on intervals $[1/2n,1/n]$ and $[1/n,3/2n]$.
Let $X = [0, 2]$ and $$f_n(x) =\begin{cases} n^2x^2 − 2nx, 0 \leq x \leq 2/n\\ 0,&2/n\lt x \leq 2\\ \end{cases}$$
Clearly $f_n $ is continuous and converges pointwise to $0$, but the convergence is not uniform.
