Generalized Minkowski inequality for $L^p$ spaces
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.