Prove sum of functions is $L^2$
By monotone convergence theorem, $$\lVert g \rVert_{L^2} = \lim_{N \to \infty}\left\lVert \sum_{k = 1}^{N}|f_n|\right\rVert_{L^2} \leq \lim_{N \to \infty}\sum_{k = 1}^{N}\lVert f_n \rVert_{L^2} = \sum_{k = 1}^{\infty}\lVert f \rVert_{L^2} < \infty.$$