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Fiji nos Jogos Olímpicos de Verão de 1968

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Fiji nos Jogos Olímpicos de Verão de 1968 Comitê Olímpico Nacional Código do COI FIJ Nome Fiji Association of Sports and National Olympic Committee site oficial (em inglês) Jogos Olímpicos de Verão de 1968 Sede Cidade do México, México Competidores 1 em 1 esporte Medalhas Pos. n/d 0 0 0 0 Participações nos Jogos Olímpicos Verão 1956 • 1960 • 1964 • 1968 • 1972 • 1976 • 1980 • 1984 • 1988 • 1992 • 1996 • 2000 • 2004 • 2008 • 2012 • 2016 Inverno 1988 • 1992 • 1994 • 1998 • 2002

If $f:Xrightarrow X$ is continuous and $(X,d)$ is compact, show there is $epsilon>0$ such that...

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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