On uniform convergence of functions with compact support

Question

If sequence of functions $f_{n}$ in $C_C(\mathbb{R})$ converges pointwise to the functions $e^{-x^2}$, then can we conclude $f_n$ converge uniformly.

Answer 1

No. Consider the sequence $$ f_n(x) = (e^{-x^2}+\chi_{[n,n+1]}), $$ where $\chi$ is the indicator function.

Notice that for any $x\in \mathbb{R}$, once $n>x$ then $f_n(x)=e^{-x^2}$, so $f_n$ converges pointwise.

On the other hand $\|f_n-e^{-x^2}\|_\infty = \|\chi_{[n,n+1]}\|_\infty = 1$ for all $n$.

I now realize you want functions in $C_C(\mathbb{R})$, but hopefully you see how to modify this example slightly to that end