Consider the following event: $S:= \{ \sup_n \ \tau_n < \infty\} $
$\tau_n- \tau_{n-1}=T_n $, where $T_n \sim exp( a_n) $
$(a_n)$ is a sequence of positive numbers in $\mathbb{R}$
Suppose: $ \sum_{n=1}^{\infty} \frac{1}{a_n} = \infty$
How can I show that $P(S)=0$?
If $\sum_{n=1}^\infty \frac1{a_n} = \infty$, then $\prod_{n=1}^\infty(1+1/a_n)=\infty$. By monotone convergence and independence, $$ \mathbb E\left[e^{-\sum_{n=1}^\infty T_n} \right] = \prod_{n=1}^\infty \mathbb E[e^{-T_n}] = \prod_{n=1}^\infty\left(1+\frac1{a_n}\right)^{-1} = 0, $$ so $$\mathbb P\left( \sum_{n=1}^\infty T_n = \infty\right) = 1. $$