Triangle Inequality for Series/Lebesgue Spaces

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^p} {\mu}$ be a sequence of $p$-integrable functions, that is, a sequence in Lebesgue $p$-space.

Suppose that for all $n \in \N$, $f_n \ge 0$ holds pointwise.

Then:


 * $\ds \norm {\sum_{n \mathop = 1}^\infty f_n}_p \le \sum_{n \mathop = 1}^\infty \norm {f_n}_p$

where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.