In the present paper the authors prove the following Theorem: Let be a nonempty closed convex subset of a real Hilbert space and be an asymptotically nonexpansive mapping (i.e. for each there exists a real number such that , and ). Suppose that the set of fixed points of , is nonempty. Let
Then the mapping on given by
has a unique fixed point in and the sequence converges strongly to the element of which is nearest to .