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  "aliases": [
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  "body": "\n## Statement\n\nLet $(X, d)$ be a nonempty [[complete metric space|08EWT7J0SC]] and let $f \\colon X \\to X$ be a [[contraction|0MT2S0ZX4Y]] with constant $k \\in [0, 1)$.\nThen $f$ has exactly one fixed point $x^*$, that is, a point with $f(x^*) = x^*$.\nMoreover, for each $x_0 \\in X$, the sequence $x_{n+1} = f(x_n)$ converges to $x^*$, and\n$$d(x_n, x^*) \\le \\frac{k^n}{1 - k} \\, d(x_1, x_0) \\quad \\text{for all } n \\ge 0.$$\n\n## Remarks\n\nProof idea: $d(x_{n+1}, x_n) \\le k^n d(x_1, x_0)$, so the sequence $(x_n)$ is a [[Cauchy sequence|0MCK8K2D7Z]] by a geometric series bound. Its limit is a fixed point because $f$ is continuous. Two fixed points $x, y$ satisfy $d(x, y) \\le k \\, d(x, y)$, so $x = y$.\nCompleteness is needed: $x \\mapsto x/2$ on $(0, 1]$ is a contraction without a fixed point.\nThe theorem gives the existence of solutions of ordinary differential equations in the Picard–Lindelöf theorem.\n\n## History\n\nStefan Banach stated the theorem in 1922.\n\n## Explanations\n\n### L1 human\n\nApply the contraction again and again, starting from any point. The points get closer to each other, and in a complete space they converge.\nThe limit does not move under $f$, and it is the only point that does not move.\n",
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