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A strictly distance-decreasing map decreases the distance between every pair of distinct points. Unlike a contraction mapping, it need not have a single Lipschitz constant below one. Two distinct fixed points would contradict strict decrease, so there is at most one. Existence can fail even on a complete metric space: on $[0,\infty )$, $T(x)=x+1/(1+x)$ has no fixed point, while the mean value theorem and $0\le T^{\prime }(x)<1$ show strict distance decrease.