# Application of Borel Cantelli lemma (almost sure convergence)

Let $$(X_n)_{n\geq 1}$$ be a sequence of real-valued random variables.

I have to proof that if for every $$\epsilon > 0: \sum_{n=1}^{\infty} \mathbb P(|X_n - X| > \epsilon) < \infty$$ , then $$X_n \to X$$ almost sure.

Here's my attempt: To prove that $$X_n$$ converges a.s. I'd use Borel Cantelli lemma. Let $$A_n = \{|X_n -X| > \epsilon\}$$ and $$\sum \epsilon^2 < \infty$$ so by Borel Cantelli lemma, $$\mathbb P(A_n$$ occurs infinitely often) = 0, then for almost all $$w$$ for all $$N$$ greater than some index $$N$$ (depending on $$w$$), $$|X_n(w) −X(w)| \leq \epsilon$$, and thus $$X_n \to X$$ a.s.

Is this OK? Any hints / help / correction is appreciated.

• Almost OK. Note that you're interested in $\mathbb P( \forall_{\varepsilon > 0} \exists_N \forall_{n \ge N} |X_n - X| \le \varepsilon\}) = 1$. You've just proved it for fixed $\varepsilon > 0$. But the point is $\mathbb P( \forall_{\varepsilon > 0} \exists_N \forall_{n \ge N} |X_n-X| \le \varepsilon\}) = \mathbb P(\{\forall_{m \in \mathbb N_+} \exists_{N } \forall_{n \ge N} |X_n - X| \le \frac{1}{m}\}) = \mathbb P(\bigcap_m \{ \exists_N \forall_{n \ge N} |X_n - X| \le \frac{1}{m}\})$. And the latter is $1$ as a countable intersection of events of probability $1$ (which you've proved) May 20, 2021 at 19:53
• Oh thank you, you're right. May 21, 2021 at 6:04