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  "body": "\n## Statement\n\nLet $X, X_1, X_2, \\dots$ be real random variables on one probability space.\nIf $X_n \\to X$ [[in probability|0EFDDXTVD6]], then $X_n \\to X$ [[in distribution|0JDAQHWA9N]].\n\n## Remarks\n\nProof: for $\\varepsilon \u003e 0$, $F_n(x) \\le F(x + \\varepsilon) + P(|X_n - X| \u003e \\varepsilon)$ and $F(x - \\varepsilon) \\le F_n(x) + P(|X_n - X| \u003e \\varepsilon)$.\nLet $n \\to \\infty$ and then $\\varepsilon \\to 0$. At a continuity point $x$ of $F$, this gives $F_n(x) \\to F(x)$.\nThe converse is false. Let $X \\sim \\mathcal{N}(0, 1)$ and $X_n = -X$ for all $n$. Then $X_n \\to X$ in distribution, but $P(|X_n - X| \u003e 1) = P(|X| \u003e 1/2)$ does not tend to $0$.\nThe converse holds when the limit is a constant: if $X_n \\to c$ in distribution for a real $c$, then $X_n \\to c$ in probability.\n\n## Explanations\n\n### L1 human\n\nIf $X_n$ is very likely to be close to $X$, then the probabilities for $X_n$ are close to those for $X$.\nBut two variables can have the same law and still be far apart, so the converse fails.\n",
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