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  "body": "\n## Proof\n\n1. Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\\Omega, \\mathcal{F}, P)$, with [[cumulative distribution function|0MXMS3AE13]] $F_X(t) = P(X \\le t)$.\n2. For each integer $n \\ge 1$ let $A_n = \\set{X \\le n}$. It is an event, because $(-\\infty, n]$ is a Borel set, and $P(A_n) = F_X(n)$.\n3. The events increase: if $X(\\omega) \\le n$, then $X(\\omega) \\le n + 1$. So $A_n \\subseteq A_{n+1}$.\n4. Their union is $\\Omega$. Let $\\omega \\in \\Omega$. The value $X(\\omega)$ is a real number, so by the [[Archimedean property|0A3GWZ86AY]] there is $n \\ge 1$ with $n \u003e X(\\omega)$. Then $\\omega \\in A_n$.\n5. By [[continuity from below|0X8HBKCSYB]], $F_X(n) = P(A_n) \\to P(\\Omega) = 1$ as $n \\to \\infty$.\n6. Let $\\varepsilon \u003e 0$. By [[the ε-N form of the limit|09G6KHS6GY]] and step 5, there is an integer $N \\ge 1$ with $\\abs{F_X(N) - 1} \u003c \\varepsilon$, that is, $F_X(N) \u003e 1 - \\varepsilon$.\n7. Let $x \\ge N$. The function $F_X$ is [[nondecreasing|0XD22PBMWA]], so $F_X(x) \\ge F_X(N) \u003e 1 - \\varepsilon$. Also $F_X(x) \\le 1$, because $F_X(x)$ is a probability.\n8. So $\\abs{F_X(x) - 1} \u003c \\varepsilon$ for all $x \\ge N$. Since $\\varepsilon \u003e 0$ was arbitrary, $\\lim_{x \\to +\\infty} F_X(x) = 1$.\n\n## Idea\n\nThe events $\\set{X \\le n}$ grow and eventually catch every outcome, because $X$ takes only real values. Continuity from below makes their probabilities tend to $1$. Monotonicity of $F_X$ then carries the limit from the integers to all real $x$.\n",
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