I'm working on the following exercise:
Let $X_1, X_2, \ldots$ be i.i.d. nonnegative random variables. By virtue of the Borel-Cantelli lemma, show that for every $c \in (0,1)$, $$ \sum_{n=1}^\infty e^{X_n} c^n \begin{cases} < \infty \textrm{ a.s.} & \textrm{if } \mathbb E[X_1] < \infty; \\ = \infty \textrm{ a.s.} & \textrm{if } \mathbb E[X_1] = \infty \end{cases} $$
I'm trying to show $\sum_{n=1}^\infty \mathbb P\left[\sum_{k=1}^n e^{X_k} c^k \geq M\right] < \infty$ for some large $M > 0$. For then, Borel-Cantelli gives us that $$ \mathbb P\left[\limsup \left\{ \sum_{k=1}^n e^{X_k} c^k \geq M\right\}\right] = \mathbb P\left[\sum_{k=1}^\infty e^{X_k} c^k \geq M\right] = 0$$ and we're done. But I don't know how to show $\sum_{n=1}^\infty \mathbb P\left[\sum_{k=1}^n e^{X_k} c^k \geq M\right] < \infty$. Any suggestions?