= Fixed point theorem for a strictly distance-decreasing map on a compact metric space
Let $X$ be a nonempty <compact space>[compact] <metric space> and let $f:X\to X$ satisfy $d(f(x),f(y))<d(x,y)$ whenever $x\neq y$. Then $f$ has exactly one <fixed point>. The distance condition implies <continuity>, and $x\mapsto d(f(x),x)$ attains a minimum by <compactness>. If a minimizer $p$ were not a <fixed point>, then $d(f^2(p),f(p))<d(f(p),p)$ would contradict minimality. Two distinct <fixed point>[fixed points] would contradict the same strict distance inequality directly. A uniform <Lipschitz continuity>[Lipschitz] constant below $1$ is unnecessary; <compactness> replaces the quantitative hypothesis of the <contraction mapping theorem>.
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