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  "body": "\n## Statement\n\nFor each $n \\ge 1$, let $a_n \\le b_n$ be real numbers and let $I_n = [a_n, b_n]$ be the closed [[interval|0TK8Y3J5RW]].\nAssume that $I_{n+1} \\subseteq I_n$ for all $n \\ge 1$.\nThen the intersection $\\bigcap_{n \\ge 1} I_n$ is not empty.\nIf moreover $b_n - a_n \\to 0$, then the intersection consists of exactly one point.\n\n## Remarks\n\nThe left endpoints $a_n$ increase and are bounded above by $b_1$.\nBy the [[completeness axiom|0AXWET80YW]], they have a [[supremum|0BTMQ04556]] $s$.\nEach $b_m$ is an upper bound of all $a_n$, so $a_n \\le s \\le b_m$ for all $n$ and $m$, and $s$ lies in every $I_n$.\nIf two points $s \u003c t$ lie in every $I_n$, then $b_n - a_n \\ge t - s \u003e 0$ for all $n$, so the lengths cannot tend to $0$.\n\nEach hypothesis is needed.\nThe open intervals $(0, 1/n)$ are nested and nonempty, but no real number lies in all of them.\nThe closed unbounded sets $[n, \\infty)$ are nested, but their intersection is empty.\nOver $\\Q$, the intervals $\\{q \\in \\Q : a_n \\le q \\le b_n\\}$ with $a_n, b_n$ the decimal approximations of $\\sqrt 2$ from below and above have empty intersection.\n\nThe theorem is the case $X = \\R$ of [[Cantor's intersection theorem|0EZBG8SKJM]].\n\n## Explanations\n\n### L1 human\n\nTake a closed interval, then a smaller closed interval inside it, then a smaller one inside that, and so on forever.\nSome point stays inside all of them.\nIf the lengths shrink to zero, the intervals close in on a single point.\nFor example, the intervals $[0, 1/n]$ all contain $0$, and $0$ is the only common point.\n\n### L2 human\n\nThe theorem is one form of the [[completeness of the real numbers|E3CPZNJZNG]], and in an Archimedean ordered field it is equivalent to the least-upper-bound property.\nIt drives bisection proofs: halve an interval, keep a half with a desired property, and repeat.\nThis is how one proves the Bolzano–Weierstrass theorem and the intermediate value theorem, and how Cantor showed that $\\R$ is uncountable.\n",
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