Does $L^2$-convergence of test functions imply a.e. convergence?

Question

Suppose I have a sequence $f_n\in C_c^\infty(\mathbb{R})$ of compactly supported smooth functions on $\mathbb{R}$ (with values in $\mathbb{R}$) that converge in the $L^2$-norm to $f\in L^2(\mathbb{R})$. Then does $f_n$ converge pointwise almost-everywhere to $f$?

(Actually I have in mind the case when $L^2(\mathbb{R})$ is replaced by $H^2(M)$, the second Sobolev space on a non-compact manifold, but I thought that perhaps this would follow from the simpler case.)

Answer 1

No always, but we can find a subsequence $(f_{n_{k}})$ that converges to $f$ pointwise a.e.

Answer 2

No, in general this is false. Luckily though there exist subsequences which converge a.e. pointwise.

See for instance this question Lp convergence and pointwise convergence. The countexample in there is given in terms of characteristic functions, you could get one for smooth functions by approximating those.