Can someone give me an example of a sequence $\{f_k\}_{k \in \mathbb N}$ of (preferably continuous) functions from any subset of $\mathbb R$ to $\mathbb R$ that converges to a function in the $L^∞$ sense but not in the $L^2$ sense? That is, as $k \rightarrow \infty$, there exists an $f$ such that
$$ \sup_{x \in \mathrm{dom} (f)} |f(x) - f_k(x)| \rightarrow 0, $$
but no $g$ such that
$$ \int_{\mathrm{dom} (g)} (g(x) - f_k(x))^2\mathop{dx} \rightarrow 0. $$
This is probably very easy, but I can't find any examples after lots of searching. Most resources say that $L^\infty$ convergence implies $L^2$ convergence in finite measure spaces, but they don't provide an example of it not applying in other spaces.
Edit: Ah, whoops, I swapped around $L^\infty$ and $L^2$!
If $f$ is any non-zero continuous function such that $\int |f(x)|^{2}dx<\infty$ then $(f(nx)) \to 0$ in $L^{2}$ but not in $L^{\infty}$.
Note that $\int |f(nx)|^{2}dx=\frac 1n \int |f(y)|^{2} dy \to 0$.
EDIT: For the revised version take $f_n(x)=\frac 1 {\sqrt n} I_{(0,n)}$. Then $f_n \to 0$ in $L^{\infty}$ but $\int |f_n|^{2}=1$ for all $n$.