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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nA map $f \\colon X \\to X$ is a contraction when there is a real number $k$ with $0 \\le k \u003c 1$ such that\n$$d\\bigl(f(x), f(y)\\bigr) \\le k \\, d(x, y) \\quad \\text{for all } x, y \\in X.$$\n\n## Remarks\n\nA contraction is a [[Lipschitz map|0TMZ21WKCN]] with a constant less than $1$. Thus it is uniformly continuous.\nThe map $x \\mapsto x/2$ on $\\mathbb{R}$ is a contraction with $k = 1/2$.\nThe condition $d(f(x), f(y)) \u003c d(x, y)$ for all $x \\neq y$ is weaker. The map $x \\mapsto x + 1/x$ on $[1, \\infty)$ satisfies it, but it is not a contraction and has no fixed point.\n\n## Explanations\n\n### L1 human\n\nA contraction pulls the points of the space closer together, each time by at least the same factor.\nIf you apply it again and again, all points are pulled toward one point.\n",
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