# Generalized Minkowski inequality for $L^p$ spaces

Given $$f_k \in L^p(\Omega)$$, I have to prove that $$\left\|\sum_{n\,=\,1}^\infty f_n\right\|_{L^p(\Omega)}\leqslant \sum_{n\,=\,1}^\infty \left\| f_n \right\|_{L^p(\Omega)}$$, where $$\left\|f \right\| \doteq \left(\displaystyle\int_\Omega \left| f\right|^p\right)^{1/p}$$ denotes the $$p$$-norm, with $$1 \leqslant p < \infty$$. Since I've already proved the original Minkowski inequality, a simple induction estabilishes the finite case, i.e., $$\left\|\sum_{n\,=\,1}^m f_n\right\|_{L^p(\Omega)}\leqslant \sum_{n\,=\,1}^m \left\| f_n \right\|_{L^p(\Omega)} (*)$$, $$\mathbb{N} \ni m < \infty$$.
So, what I've thought is, since $$L^p(\Omega)$$ is a Banach space, to suppose that $$\sum_{n\,=\,1}^\infty \left\| f_n \right\|_{L^p(\Omega)} < \infty$$, which would give us that $$\sum_{n\,=\,1}^\infty f_n$$ converges in $$L^p(\Omega)$$, by completeness. Hence, the desired inequality follow by making $$m \longrightarrow \infty$$ in $$(*)$$, but I'm not completely sure about this.
Any help will be appreciated.

There is really nothing difficult. What you know is that $$g_n\sum_{i=1}^n f_n$$ converges to some element $$g$$ in $$L^p$$. In particular, $$\|g_n\| \to \|g\|$$. Since
$$\| g_n\| = \left\| \sum_{i=1}^n f_n \right\|\le \sum_{i=1}^n \|f_n\| \le \sum_{i=1}^\infty \|f_n\|$$
The sequence of real numbers $$(\|g_n\|)_{n=1}^\infty$$ is uniformly bounded by $$\sum_{i=1}^\infty \|f_n\|$$, this implies $$\|g\| \le \sum_{i=1}^\infty \|f_n\|.$$
Note that $$g = \sum_{i=1}^n f_n$$. Thus you are done.