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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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    Solution Preview

    ** 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 ...

    Solution Summary

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