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  "body": "\n## Definition\n\nLet $(a_n)$ be a [[real sequence|0CTV783HYT]].\nThe sequence is contractive if there is a real number $C$ with $0 \u003c C \u003c 1$ such that\n$$|a_{n+2} - a_{n+1}| \\le C\\, |a_{n+1} - a_n| \\quad \\text{for all } n \\in \\mathbb{N}.$$\n\n## Remarks\n\nEach contractive sequence is a [[Cauchy sequence|0QZMTN5AXX]]. By induction $|a_{n+1} - a_n| \\le C^n |a_1 - a_0|$, and the [[geometric series|01X5CVV1Q5]] bounds $|a_m - a_n|$. So each contractive sequence converges.\nIf $f \\colon \\mathbb{R} \\to \\mathbb{R}$ is a contraction mapping, so that $|f(x) - f(y)| \\le C|x - y|$ with $C \u003c 1$, then each sequence $a_{n+1} = f(a_n)$ is contractive.\nIt is not enough that $|a_{n+2} - a_{n+1}| \u003c |a_{n+1} - a_n|$ for all $n$. The partial sums of the harmonic series satisfy this, and they diverge.\n\n## Explanations\n\n### L1 human\n\nIn a contractive sequence, each step is at most a fixed fraction of the step before.\nThe steps shrink like a geometric sequence, so the total distance travelled is finite, and the sequence converges.\n",
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