Formalities about the expectation random variable

Question

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of non-negative random variables, so all $\mathbb{E}[X_n]\leq \infty$ are well-defined. Then $\sum_{n \in \mathbb{N}} X_n$ is a well-defined random variables as well (why?), non-negative and by Beppo-Levi and the linearity of integrals, we have that \begin{align} \mathbb{E}\big[\sum_{n \in \mathbb{N}} X_n\big] = \sum_{n \in \mathbb{N}}\mathbb{E}[ X_n ]. \end{align} Moreover, I am wondering why $\sum_{n \in \mathbb{N}}\mathbb{E}[ X_n ] < \infty $ implies that $\sum_{n \in \mathbb{N}} X_n < \infty $ a.s.?