I would like to show that
If $E(X_n) = 1/n$ and $\mathrm{Var}(X_n) = 1/n^2$ then $X_n\to0$ almost surely.
I can not find some relation between the almost sure convergence and $E(X_n)$ and $Var(X_n)$.
2026-03-27 00:59:55.1774573195
If $E(X_n) = 1/n$ and $\mathrm{Var}(X_n) = 1/n^2$ then $X_n\to0$ almost surely
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HINT
Chebyshev gives $P(|X_n-1/n| \ge \epsilon )\le \frac{1}{\epsilon^2n^2}.$ Then apply Borel-Cantelli. The key relationship here is that often "in probability" + "Borel-Cantelli" = "almost sure"