Let \(S\) be the usual class of normalized univalent functions in the unit disk \(\Delta\). Given \(M>1\) let \(S(M)= \{f\in S: | f(z)|< M\) for all \(z\in\Delta\}\) and let \(M(S)= \{f\in S: f(z)/M\in f(\Delta)\) for all \(z\in \Delta\}\). The author calls such functions \(M\)-almost starlike and proves that \(\varphi\in S\) is in \(M(S)\) if and only if there is a function \(f\in S(M)\) such that the sequence \(\{f_ n\}\) defined by \(f_ 1= f\), and \(f_{n+1}= Mf_ n(f/M)\), converges to \(\varphi\). The reviewer fails to see much value in studying this class, but the lemmas and their proofs are rather interesting. The tools used are quite elementary, but are put together in a very clever way. It is years since the reviewer has seen a proof making use of the sequence-subsequence lemma (If a sequence of functions \(\{f_ n\}\) is such that every subsequence has itself a subsequence which converges to the same function \(g\), then \(f_ n\to g\)).