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  "body": "\n## Statement\n\nLet $X$ be a real random variable with [[cumulative distribution function|0MXMS3AE13]] $F_X$.\nThen\n$$\\lim_{x \\to +\\infty} F_X(x) = 1.$$\n\n## Remarks\n\nThe proof: the events $\\{X \\le n\\}$ increase with $n$ and their union is $\\Omega$. So [[continuity from below|0X8HBKCSYB]] gives $F_X(n) \\to 1$, and $F_X$ is nondecreasing.\nIn the same way, $\\lim_{x \\to -\\infty} F_X(x) = 0$, because the events $\\{X \\le -n\\}$ decrease to $\\varnothing$.\nThe proof uses that $X$ takes only real values. For a random variable with values in $[-\\infty, +\\infty]$, the limit is $1 - P(X = +\\infty)$.\n\n## Explanations\n\n### L1 human\n\nFor a very large $x$, it is almost sure that $X \\le x$, because $X$ is some real number.\nSo the probability $F_X(x)$ gets as close to $1$ as we like.\n",
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