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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. Let $t \\in \\R$. The complement of $(-\\infty, t]$ is $(t, \\infty)$, which is [[open|01ZQQDCERP]]. So $(-\\infty, t]$ is [[closed|0G5AV9Q4RR]].\n3. The [[Borel σ-algebra|0BVHZXWKCD]] contains the open set $(t, \\infty)$ and is closed under complements. So it contains $(-\\infty, t]$.\n4. Since $X$ is a random variable, $\\set{X \\le t} = X^{-1}((-\\infty, t])$ is in $\\mathcal{F}$.\n5. Let $x \\le y$ be real numbers. If $\\omega \\in \\Omega$ satisfies $X(\\omega) \\le x$, then $X(\\omega) \\le y$. So $\\set{X \\le x} \\subseteq \\set{X \\le y}$.\n6. By step 4 both sets are events. By [[monotonicity of probability|0464Z0JP2E]], $P(X \\le x) \\le P(X \\le y)$, that is, $F_X(x) \\le F_X(y)$.\n\n## Idea\n\nRaising the threshold from $x$ to $y$ can only enlarge the event $\\set{X \\le x}$. A larger event has at least the same probability.\n",
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