Let $X_i$ be $\lambda_i$ Poisson-distributed random variables, where $0<\lambda_i\le\Lambda$ for some finite $\Lambda$. I want to show that $$P(X_n\ge n \text{ for infinitely many n})=0$$ using Borel-Cantelli. To this end, I need to show that $$\sum_{j=1}^\infty P(A_j)=\sum_{j=1}^\infty P(X_j\ge j)<\infty$$ Direct computation gives us $$\sum_{j=1}^\infty\sum_{k=j}^\infty \frac{\lambda_j}{k!}e^{-\lambda_j}$$ But I don't know how to proceed.
Can we simply calculate the sum directly? Is there a smarter way to show the series converges?
Edit: It is clear that the series terms converge to $0$, but I don't have an idea for a ratio or bound test.
Using $0 < \lambda_i \leq \Lambda$ the sum can be bounded as
$$\sum_{j=1}^{\infty} \sum_{k=j}^{\infty} \frac{ \lambda_j}{k!} e^{-\lambda_j} \leq \Lambda \sum_{j=1}^{\infty} \sum_{k=j}^{\infty} \frac{1}{k!} = \Lambda \sum_{n=1}^{\infty} \frac{n}{n!} = \Lambda e$$