I am relatively new to Fourier series. Let $f \in L^2([-\pi, \pi])$ with $f(-\pi) = f(\pi)$. The Fourier coefficients of $f$ are given by \begin{align*} \hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int} \, dt, \end{align*} the partial sums of the Fourier series of $f$ are given by \begin{align*} s_N(t) = \sum \limits_{-N}^N \hat{f}(n) e^{int}, \end{align*} and the Fourier series of $f$ is given by \begin{align*} F(t) = \sum_{-\infty}^\infty \hat{f}(n) e^{int}. \end{align*}
My question is: do the functions $s_N$ converge pointwise to $F$?
It seems to me that this should both be true, but then I derive a contradiction. It is known that the $s_N$ converge to $f$ in the $L^2$ norm, and so a subsequence $(s_{N_k})_k$ converges pointwise to $f$. This implies that $f=F$ almost everywhere, but since $s_N \rightarrow F$ pointwise, we get $s_N \rightarrow f$ pointwise almost everywhere.
But as far as I can tell, pointwise convergence of the Fourier series to the original function $f$ is a very difficult question. Is the "almost everywhere" what makes a difference? Did I make a mistake in my argument? Or do the $s_N$ just not converge pointwise to $F$?