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Let $(X,d)$ be a complete metric space and $$f \ :\ X\rightarrow X$$ be a map such that , for some positive integer $k$ , $$f\circ f\circ .....\circ f(\ k\ \ fold\ \ composition\ \ with\ \ itself\ )$$ is a contraction.
Then $f$ has a unique fixed point .
How should I approach this problem $?$
Sorry for the lack of efforts here .
Please give me some hints as to how to begin the thinking $?$