Why doesn't proving $f\in L^2(S^1)$ can be expanded into a Fourier series imply pointwise convergence of continuous functions?
In the book I'm studying it is stated that the unit-perpendicular family $\{e_n\}_{n=1}^\infty$, where $e_n = e^{2\pi i n x}$ and $n\in \mathbb{Z}$, is a basis for $L^2(S^1)$ meaning $f\in L^2(S^1)$ can be expanded into Fourier series (where $S^1$ corresponds to the interval $0\leq x \leq 1$ with the endpoints set equal).
It is straight-forward to prove $n\in \mathbb{Z}$ is a unit-perpendicular. It remains to show that $\{e_n\}$ spans $L^2(S^1)$: to do so the author prove that for any $f\in C^p(S^1)$, where $1\leq p < \infty$ is the number of times the functions is differentiable, we get that the partial sums
$S_n = \sum_{|k|\leq n}\hat{f}(k)e_k$,
where $\hat{f}(k)$ is a Fourier coefficient, converges uniformly to $f$. It's left as an exersise to complete the proof, with the hint that $C^p(S^1)$ is dense in $L^2(S^1)$ which I interpret as we should use the result for smooth functions to show that $\{e_n\}$ spans $L^2(S^1)$. For any $g \in L^2(S^1)$ this seems to follow by the inequality
$||g-\sum_{|k|\leq n}\hat{f}_m(k)e_k||_2\leq||g-f_m||_2 + ||f_m-\sum_{|k|\leq n}\hat{f}_m(k)e_k||_2$,
where $f_m=\lim_{n\rightarrow \infty}\sum_{|k|\leq n}\hat{f}_m(k)e_k$ and $f_m$ is the smooth function we use to approximate arbitrary $f\in L^2(S^1)$. You may please confirm if this is a correct solution to the exercise. But my main concern is why this doesn't mean that we also have pointwise convergence of (non-differentible) continuous functions considering we can then expand them into Fourier series which would appear to imply pointwise convergence of the partial sums. The reason I'm asking is that I know that (almost everywhere) pointwise convergence must be proved separately and that this is a complicated task (Carleson theorem).