The sequence Txn so constructed contains a subsequence TXnk which converges to some x X and which, by Ascoli's theorem [3], is equicontinuous. It follows in turn from (1) that the sequence Xnk is equicontinuous. To complete the proof it suffices to show xn,(t)-x(t) in B. For then the sequence Xnk is compact, again by Ascoli's theorem, so that Xn,kX; hence Txnk->Tx, which together with Txnk-)x yields Tx=x. That indeed Xnk(t)->x(t) is immediate from the fact that xnk(0) =u(0) =x(0) and that, for t50 and sufficiently large k,