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Euclidean metric proof

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Let d be the usual Euclidean metric on R^n and f:R^n -> R^n be any function satisfying d(f(x),f(y))<d(x,y) for all distinct x,y in R^n. If B is a bounded subset of R^n such that f(B) is contained in B, show there is a unique b in the closure of B such that f(b) = b. I can show the uniqueness of such a b. Please give a detailed solution of the existence.

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This solution provides an example of proving the existence of a unique element in a closure.

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** Please see the attached file for the complete solution response **

First, f is continuous: if (please see the attached file) then (please see the attached file).

Let (please see the attached file). Define recursively ...

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