sum of function of stopping times

Question

Let $\tau_{n}$ be a an increasing sequence of stopping times such that

$$\forall T \in [0, \infty), \ P(\lim_{n\rightarrow \infty} \tau_{n} \leq T) = 0$$

Does this imply that:

$$E\biggl(\sum_{n=1}^{\infty} e^{-\lambda \tau_{n}}I_{(\tau_{n}<\infty)}\biggr)<\infty$$

I know the opposite is true, but I am not sure about this one.

Answer 1

No. Consider the (constant) stopping times $\tau_n=\frac {\ln n}\lambda$. In this case, the hypothesis holds but your series becomes $\sum \frac 1 n$.