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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.

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    $\begingroup$ 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) $\endgroup$
    – Presage
    May 20, 2021 at 19:53
  • $\begingroup$ Oh thank you, you're right. $\endgroup$
    – Stanisla
    May 21, 2021 at 6:04

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