Convergence of series of probabilities

Question

I am trying to see the following implication: Let $(X_{n})_{n\in \mathbb{N}}$ be a sequence of independant random variables and $(c_{n})_{n\in \mathbb{N}}$ a sequence of positive reals such that $c_{n}X_{n}\to 0$ for $n\to \infty$ pointwise for all $\omega$ in a set of positive measure. Then for every constant $a>0$ $$\sum_{n=1}^{\infty}{P\left(|X_{n}|\geq \frac{a}{c_{n}}\right)}<\infty.$$ Any ideas?

Answer 1

This reiterates the converse of the Borel Cantelli Lemma. If the sum were infinite for some $a>0$, it would imply $c_n|X_n|\geq a>0$ infinitely often with probability 1, contradicting $c_nX_n\rightarrow 0$.