Math Help - Problems about probability

Let $(A_\beta)_{\beta \in B}$ be a family of pairwise disjoint events. Show that if $P(A_\beta) > 0$ for each $\beta \in B$, then $B$ is at most countable.
Let $(A_\beta)_{\beta \in B}$ be a family of pairwise disjoint events. Show that if $P(A_\beta) > 0$ for each $\beta \in B$, then $B$ is at most countable.
$B=\bigcup_{n=1}^{\infty}I_n,$ where $I_n= \left \{\beta \in B: \ \frac{1}{n+1} \leq P(A_{\beta}) \leq \frac{1}{n} \right \}.$ now if $|I_n|=\infty,$ for some $n,$ then $P(\bigcup_{\beta \in B}A_{\beta})=\infty,$ which is nonsense. so each $I_n$ is a finite set and thus $B$ is (at most) countable.