Variant of the Contraction Mapping Theorem

Question

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that

||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to itself?

Answer 1

Hint $C=[2, \infty) \subset \mathbb R$ and $$f(x)=x+\frac{1}{x}$$