Generalized Minkowski inequality for $L^p$ spaces

Question

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.

Answer 1

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.