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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \\subseteq X$.\nThe set $S$ is bounded when there is a real number $M$ such that $d(x, y) \\le M$ for all $x, y \\in S$.\n\n## Remarks\n\nThe empty set and each finite set are bounded.\nIf $X$ is not empty, then $S$ is bounded if and only if $S$ lies in some [[closed ball|05DSV40EB9]] $\\bar{B}(x, r)$ with $x \\in X$ and $r \\ge 0$.\nIn $\\mathbb{R}$ with the usual metric, $S$ is bounded in this sense if and only if $S$ is bounded above and bounded below.\nBoundedness depends on the metric: $\\mathbb{R}$ is not bounded for $|x - y|$, but it is bounded for $\\min(|x - y|, 1)$.\n\n## Explanations\n\n### L1 human\n\nA set is bounded when you can put all of it inside one large ball.\nFor example, the interval $(0, 1)$ is bounded, but the set $\\mathbb{N}$ of natural numbers is not bounded in $\\mathbb{R}$.\n",
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