A smooth sequence of function converging in the L^2 norm implies uniform convergence. I am stuck on the following.
Let $f_n$ be a sequence of functions on the circle, and suppose we know the following about its Fourier coefficients:


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*For every $n\in\mathbb{N}, $$|\hat{f}_n(k)|\leq \frac{C}{1+|k|^N}$, for every $N\in\mathbb{N}$. That is, the $f_n$ is sequence of $C^{\infty}$-functions.

*$\lim_{n\rightarrow \infty} \sum_{k\in\mathbb{Z}}|\hat{f}_n(k)|^2 = 0$, i.e.  $f_n$ converges to zero in the $L^2$-norm.


Does this imply uniform convergence to zero:
$$ ||f_n||_{\infty} := \sup_{x\in[0,2\pi)} |f(x)|  \rightarrow 0,\quad \mbox{as }n\rightarrow\infty\mbox{ ?} $$

Hint: Show the $f_n$ are equicontinuous. If we didn't have uniform convergence to $0,$ then there would exist $\epsilon>0$ and a subsequence $n_k$ such that $|f_{n_k}(t_k)| \ge \epsilon$ for all $k.$ Use equicontinuity to show that then the $L^2$ norms of $f_{n_k}$ do not converge to $0.$ 
