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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \\subseteq X$.\nThen $S$ is [[totally bounded|0WE5BBHGM6]] if and only if for each real $\\varepsilon \u003e 0$ there is a finite set $F \\subseteq S$ with\n$$S \\subseteq \\bigcup_{y \\in F} B(y, \\varepsilon).$$\n\n## Remarks\n\nSuch a finite set $F$ is an $\\varepsilon$-net of $S$. Here $B(y, \\varepsilon)$ is an [[open ball|0DC3MPR7C9]].\nProof: an $\\varepsilon$-net gives a cover by balls of diameter at most $2\\varepsilon$. Conversely, from a cover by sets $A_k$ of diameter less than $\\varepsilon$, choose one point of $S$ in each $A_k$ that meets $S$.\nThe centres can be taken in $S$ or in $X$. Both choices give the same notion.\n\n## Explanations\n\n### L1 human\n\nFor each $\\varepsilon$, you can place finitely many points in $S$ such that each point of $S$ is within $\\varepsilon$ of one of them.\nFor $[0, 1]$ and $\\varepsilon = 0.1$, the points $0, 0.1, 0.2, \\ldots, 1$ do the job.\n",
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