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  "body": "\n## Example\n\nLet $S$ be a nonempty set.\nLet $B(S)$ be the set of bounded functions $f \\colon S \\to \\mathbb{R}$, that is, the functions for which there is a real $M$ with $|f(s)| \\le M$ for all $s \\in S$.\nFor $f, g \\in B(S)$, let\n$$d_\\infty(f, g) = \\sup_{s \\in S} |f(s) - g(s)|.$$\nThen $d_\\infty$ is a [[metric|063W5JMY85]] on $B(S)$, the uniform metric.\n\n## Remarks\n\nThe [[supremum|0BTMQ04556]] is finite, because $|f(s) - g(s)| \\le |f(s)| + |g(s)|$ and both functions are bounded.\nA sequence $(f_k)$ converges to $f$ in this metric if and only if $f_k \\to f$ uniformly on $S$.\nFor $S = \\{1, \\ldots, n\\}$, it is the [[sup metric on R^n|0VTAXQEWQD]].\nIt is $d_\\infty(f, g) = \\|f - g\\|_\\infty$ for the [[sup norm|0PARCKE8WP]].\n\n## Explanations\n\n### L1 human\n\nTwo functions are close in the uniform metric when their graphs are close at every point at the same time.\nFor example, on $[0, 1]$ the distance between $f(x) = x$ and $g(x) = x^2$ is $1/4$, the largest gap, at $x = 1/2$.\n",
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