Verify that x* = limk →∞ xk as defined in the preceding proof is a fixed point of f, that is, f (x*) = x*.
Schauder's theorem is frequently applied in cases where the underlying space is not compact. The following alternative version relaxes this condition to require that the image lie in a compact set. A function f: X → Y is called compact if f(X) is contained in a compact set of Y.