Uniform convergent in $C_0$

Question

Let $f_n,f\in C_0(\mathbb{R})$ such that $f_n$ converges to $f$ pointwise. Is it always true that $f_n$ converges to $f$ uniformly, i.e. in sup norm?

Answer 1

No. Let $f$ be the null function and let$$f_n(x)=\begin{cases}\sin(x)&\text{ if }x\in\bigl[n\pi,(n+1)\pi\bigr]\\0&\text{ otherwise.}\end{cases}$$Then each $f_n$ (as well as $f$) belongs to $C_0(\mathbb R)$ and $(f_n)_{n\in\mathbb N}$ converges pointwise to $f$ and, but not uniformly, since$$(\forall n\in\mathbb N):f_n\left(n\pi+\frac\pi2\right)=\pm1.$$