I'm working on the following excersise:
Given $x\in(0,\infty)$ and $0\le\lambda_n\lt x, \forall n\in\mathbb{N}$ and the sequence of random variables $(X_n)_{n \in \mathbb{N}} \sim Poi(\lambda_n)$ show that $P[X_n \geq n \text{ for infinite number of n}] = 0$
I wanted to use the Borel-Cantelli Lemma but got stuck. I know there is an error in my calculation since I would suspect the sum being finite. A hint would be appreciated:
$$ \begin{aligned} \sum_{n=1}^{\infty} P\left(X_n \geqslant n\right) & =\sum_{n=1}^{\infty} 1-\underbrace{P\left(X_n<n\right)}_{F_{X_n}(n)} \\ & =\sum_{n=1}^{\infty} 1-\sum_{k=1}^n \frac{\lambda_n^k}{k !} e^{-\lambda_m} \\ & =\sum_{n=1}^{\infty} 1-e^{-\lambda_m} \sum_{k=1}^n \frac{\lambda_n^k}{k !} \\ & \leq \sum_{n=1}^{\infty} 1-e^{-\lambda_m} \sum_{k=1}^n \frac{x^k}{k !} \\ & \leq \sum_{n=1}^{\infty} 1-e^{-\lambda_m} \cdot e^x \\ & =\sum_{n=1}^{\infty} 1-e^{x-\lambda_m} \end{aligned} $$