Convergence in $L^p$ norm implies pointwise convergence almost everywhere? [duplicate]

Fix a real number $1\leq p<\infty$.

Is it true that if functions $f\in L^p(\mathbb{R})$ and $f_1,f_2,\ldots\in L^p(\mathbb{R})$ are such that $\|f_n-f\|_p\rightarrow 0$ as $n\rightarrow \infty$, then $f_n(y)\rightarrow f(y)$ as $n\rightarrow\infty$ for almost every $y\in\mathbb{R}$?

It really shouldn't be true, but what would be a counterexample?