Exercise: almost sure convergence

Question

Let $(X_{n}), X$ be real-valued random variables such that $X_{n} \to X$ in probability and $$ \sum_{n=1}^{\infty} P(|X_{n} - X| > a_{n}) < \infty $$ for some null sequence $(a_{n})$. Show that $X_{n} \to X$ almost surely.

We defined almost sure convergence as: $$ P(\limsup_{n \to \infty} |X_{n} - X| > 0) = 0 $$

Not sure what to do. Seems that I cant use Borel-Cantelli...

Thanks.

Answer 1

Since $a_n\to0$ as $n\to\infty$, for each $\varepsilon>0$ there exists $N(\varepsilon)$ such that $a_n<\varepsilon$ for each $n>N(\varepsilon)$. For each $\varepsilon>0$, we have that $$ \sum_{n=N(\varepsilon)+1}^\infty P(|X_n-X|>\varepsilon)\le\sum_{n=N(\varepsilon)+1}^\infty P(|X_n-X|>a_n)<\infty. $$ Hence, the Borel-Cantelli lemma shows that $X_n\to X$ almost surely as $n\to\infty$.