Proof that $ \sum_{i=1}^\infty a_n$ is converges almost surely.

Question

Let $\{a_n\} $ be a positive number sequence and $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If we have $$\sup\limits_{t>0} \left( t. \mathbb{P} \left\{ \sum_{i=1}^\infty a_n >t \right\} \right) <\infty,$$ then $ \sum_{i=1}^\infty a_n$ is converges almost surely. Can you explain for me?

Answer 1

I presume $a_n$ are random variables. Label the events A := "$\sum_n a_n$ converges" and $B_N$ := "$\sum_n a_n > N$". We know that $\mathbb{P}[B_N] \leq C / N$ for all $N \geq 1$ for some $C \geq 0$. Note that $B_{N+1} \subset B_N$. Now, $\mathbb{P}[A^c] = \mathbb{P}[\bigcap_{N \geq M} B_N] \leq \mathbb{P}[B_M] \leq C / M$ for any $M \geq 1$. Taking $M \to \infty$ shows $\mathbb{P}[A^c] = 0$. Therefore $\mathbb{P}[A] = 1$. In other words, "$\sum_n a_n$ converges" almost surely.