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2 $begingroup$ If $f:Xrightarrow X$ is a continuous function with no fixed points and $(X,d)$ is compact, show there is some $epsilon>0$ such that $d(x,f(x))>epsilon$ for all $xin X$ . My hope is to argue as follows, but I seem to be ending up with a slightly stronger statement which is what worries me a bit. Since $X$ is compact we know that $Xtimes f(X)$ is compact and therefore $d: Xtimes f(X)rightarrow (0,infty)$ has compact image. So $d$ attains a minimum, thus there is some $epsilon>0$ for which $d(x,f(x))geq epsilon$ for all $xin X$ . Does this seem ok? general-topology proof-verification metric-spaces compactness share | cite | improve this question ...