Example of a pointwise convergent functional sequence that is not compactly convergent.

Question

I'm looking for an example of a pointwisely convergent functional sequence $\{f_n\}_{n \geq 0}$ (where $f_n:\mathbb{R}\to\mathbb{R}$) that is not compactly convergent.

I'm not sure if it is even possible since all of the sequences I come up with are at least compactly convergent if if they are pointwisely convergent:

1. $f_n(x)=\dfrac{x}{n}$ is not uniformly convergent, but it is compactly convergent.
2. $f_n(x)=\sum\limits_{k=1}^{n}\dfrac{e^x}{e^k}$ doesn't fit my conditions as well.

Could you please give me a hint how I can get such a sequence?

Answer 1

Take odd $f_n$s with

$$f_n(x) = \begin{cases} nx, & \text{if $0\le x < 1/n$} \\ 2-nx, & \text{if $1/n\le x \le 2/n$.} \\ 0, & \text{if x > 2/n} \end{cases}$$

$f_n(x)\to 0$ pointwisely, since for any $x$ there is some $N_x$ that $x\in[-2/n,2/n]^c$ (namely $f_n(x)=0$) for any $n\ge N_x$. However the convergence is not compactly since $\max_{|x|\le1}|f_n(x)| = 1$ for any natural $n$.