# If the infinite sum of the norms $||f_n||$ converges then $\sum f_n$ converges to some $f \in \mathcal{L}_2(\Omega)$

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 $$\{|\sum_{n=1}^k f_n(x)-f(x)|^2\}_{k=1}^\infty$$.

Let $$S_N = \sum_{n \leq N} |f_n|$$. By construction you have that $$\{S_N\}$$ is a Cauchy sequence in $$L^2$$; indeed for any $$\epsilon > 0$$ there is a $$N_0(\epsilon)$$ sufficiently large such that for $$N \geq M \geq N_0(\epsilon)$$, you have $$\|S_N - S_M\|_2 \leq \sum_{N < j \leq M} \|f_n\|_2 \leq \sum_{j > N_0(\epsilon)} \|f_j\|_2 < \epsilon.$$
Since $$L^2$$ is complete, the sequence $$S_N$$ converges to a limit $$S$$ in $$L^2$$, and consequently there is a subsequence of $$S_N$$ which converges pointwise almost everywhere to $$S$$. However, since for such $$x$$, the sequence $$S_N(x)$$ is monotone, this implies that except for a set of measure zero, $$S_N$$ converges pointwise to $$S$$.
Now note that for those $$x$$ such that $$S_N(x)$$ converges, you must have that $$\sum_{n \geq 1} f_n(x)$$ converges.