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  "body": "\n## Statement\n\nLet $X, X_1, X_2, \\dots$ be real random variables on one probability space.\nIf $X_n \\to X$ [[almost surely|04P7D63W8P]], then $X_n \\to X$ [[in probability|0EFDDXTVD6]].\n\n## Remarks\n\nProof: fix $\\varepsilon \u003e 0$ and let $A_n = \\bigcup_{m \\ge n} \\{|X_m - X| \u003e \\varepsilon\\}$. The events $A_n$ decrease, and their intersection has probability $0$ by almost sure convergence.\nBy the [[continuity of probability|0X8HBKCSYB]], $P(A_n) \\to 0$. Since $\\{|X_n - X| \u003e \\varepsilon\\} \\subseteq A_n$, the claim follows.\nThe converse is false. Let $X_n$ be independent with $P(X_n = 1) = 1/n$ and $P(X_n = 0) = 1 - 1/n$. Then $X_n \\to 0$ in probability.\nBut by the second Borel–Cantelli lemma, $X_n = 1$ for infinitely many $n$ almost surely, so $X_n$ does not converge to $0$ almost surely.\n\n## Explanations\n\n### L1 human\n\nIf almost every run of the experiment gives a sequence that settles down at $X$, then for large $n$ a large deviation becomes unlikely.\n",
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