I have to prove this theorem:
Let $\{f_n\}_n\subset \mathcal{L}_2(\Omega)$ such that $$\sum_{n=1}^\infty ||f_n|| < +\infty$$
Then, $\sum_{n=1}^\infty f_n$ converges pointwise almost anywhere (i.e. possibly not in a null set) to some function $f \in \mathcal{L}_2(\Omega)$. Moreover, the sum converges to $f$ in $\mathcal{L}_2(\Omega)$. (Note that the first convergence is pointwise and the second is L2 convergence)
I have already shown that $\{S_k^2\}_k$ converges to some Lebesgue integrable function $g$ where $$S_k(x):=\sum_{n=1}^k|f_n(x)|$$ for $x\in\Omega$.
Now I want to deduce that $\sum_{n=1}^\infty f_n$ converges almost everywhere to some $f$ and then prove that $|f|^2$ is Lebesgue integral so then $f\in \mathcal{L}_2(\Omega)$, but I don't know where to start with this step.
Then I would just use dominated convergence theorem for $\{|\sum_{n=1}^k f_n(x)-f(x)|^2\}_{k=1}^\infty$.