Given a normed vector space X, a self-mapping P of X is said to be asymptotically regular if for every point x of X we have \(\| P^{n+1}x-P^ nx\| \to 0\) as \(n\to \infty\). First, a new and simple proof is given to the theorem of S. Ishikawa, namely: Theorem B: Let D be a subset of a normed space X and \(T: X\to X\) be a nonexpansive mapping. Given a sequence \((x_ n)\) in D and a sequence \((t_ n)\) of real numbers satisfying (i) \(0\leq t_ n\leq t<1\) and \(\sum^{\infty}_{1}t_ n=\infty,\) (ii) \(x_{n+1}=(1-t_ n)x_ n+t_ nTx_ n\) for \(n=1,2,...,\) if \((x_ n)\) is bounded then \(\| Tx_ n-x_ n\| \to 0\) as \(n\to \infty.\) This theorem is then extended to multivalued mappings which are nonexpansive in the sense defined by S. Nadler. Finally a sufficient condition is presented under which the identity mapping of X can be replaced by a nonexpansive and asymptotically regular mapping. All the proofs of these results are simply based on an elementary lemma on convergent sequences in normed vector spaces which in itself is also interesting.