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  "body": "\n## Statement\n\nLet $(a_n)$ be an increasing real sequence, in the sense of a [[monotone sequence|0KN22AAW5R]].\nAssume that $(a_n)$ is bounded above.\nThen $(a_n)$ converges, and its limit is the supremum $\\sup \\{a_n : n \\in \\mathbb{N}\\}$.\n\n## Remarks\n\nThe supremum $s$ exists by the completeness of $\\mathbb{R}$. For $\\varepsilon \u003e 0$ there is $N$ with $a_N \u003e s - \\varepsilon$, and then $s - \\varepsilon \u003c a_n \\le s$ for all $n \\ge N$.\nThe statement fails in $\\mathbb{Q}$. The decimal approximations $1, 1.4, 1.41, 1.414, \\dots$ of $\\sqrt{2}$ increase and are bounded, but they have no limit in $\\mathbb{Q}$.\nThe node [[decreasing sequence bounded below converges|02V2GYAC8P]] gives the dual statement.\nAn increasing sequence that is not bounded above [[tends to infinity|09AZCV6GMC]].\n\n## Explanations\n\n### L1 human\n\nAn increasing sequence that cannot pass a ceiling must pile up somewhere.\nIt piles up at the lowest ceiling, which is the supremum of its terms.\n",
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