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  "body": "\n## Statement\n\nLet $F$ be an [[Archimedean ordered field|0HQARMKEV5]].\nThen the following five conditions on $F$ are equivalent.\n\n1. Each nonempty subset of $F$ that is bounded above has a supremum in $F$ (the [[least-upper-bound property|0AXWET80YW]]).\n2. Each [[Cauchy sequence|0QZMTN5AXX]] in $F$ converges in $F$.\n3. For each sequence of closed bounded intervals $[a_1, b_1] \\supseteq [a_2, b_2] \\supseteq \\cdots$ in $F$, some point of $F$ lies in all of them (the [[nested intervals property|MF56GK7QY2]]).\n4. Each bounded [[monotone sequence|0KN22AAW5R]] in $F$ converges in $F$.\n5. Each bounded sequence in $F$ has a subsequence that converges in $F$.\n\nThe field $\\R$ of [[real numbers|08J2M1JYAG]] satisfies all five conditions.\n\n## Remarks\n\nThe standard proof goes in a cycle.\nFrom (1), a bounded increasing sequence converges to the supremum of its terms, which gives (4).\nFrom (4), each sequence has a monotone subsequence, which gives (5).\nA Cauchy sequence is bounded, and if a subsequence converges then the whole sequence converges, so (5) gives (2).\nFrom (2), repeated bisection between a point of a set and an upper bound gives a Cauchy sequence.\nIts limit is the supremum, and the Archimedean property makes the interval lengths tend to $0$.\nFor (3), the left endpoints of nested closed intervals have a supremum under (1), and this supremum lies in every interval.\nConversely, (3) gives (1) by the same bisection, with the Archimedean property again making the lengths tend to $0$.\nThe field $\\Q$ satisfies none of the conditions: the decimal approximations $1, 1.4, 1.41, 1.414, \\ldots$ of $\\sqrt 2$ form a Cauchy sequence with no limit in $\\Q$.\nThe Archimedean hypothesis matters for (2): there are non-Archimedean ordered fields in which each Cauchy sequence converges, but which lack the least-upper-bound property.\nConditions (1), (4) and (5) each imply the Archimedean property by themselves.\nUp to isomorphism, $\\R$ is the only [[complete ordered field|0A9E6X0S3E]].\n\n## Explanations\n\n### L1 human\n\nThe real line has no gaps, and there are several ways to say this.\nA bounded set has a least upper bound; a sequence whose terms crowd together has a limit; shrinking closed intervals always share a point.\nThese sound different, but each one implies the others.\nThe rational numbers fail all of them, because $\\sqrt 2$ is missing.\n\n### L2 human\n\nIn a course, one of the conditions is taken as an axiom or proved from a construction of $\\R$, and the others follow as theorems.\nDedekind cuts give the least-upper-bound property directly, while Cauchy sequences give condition (2) directly.\nThe [[Bolzano–Weierstrass theorem|0KDJD249DT]] is condition (5), and the [[monotone convergence theorem|0FM6N4R0VB]] is condition (4).\nThe intermediate value theorem and the extreme value theorem both rest on completeness, and both fail over $\\Q$.\n",
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