About the sequence x kEZ we have the following a x 1 Bxo b for each t 10 b t b MKk b x 0 x 1 k 1 11 b t b k20 c x k Z is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8 B x b n which is given by x t lim x 1 tet b t b Thus x 1 lim x 1 Finally the validity of this integral equation has the following two implications X t x f s x s ds x 0 x x lim ff s x s ds x lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function rx 7 satisfying the integral equation x 1 x f f s x s ds for all 1 t b t b 2 f s x s ds ff s x s ds f 1 X 1 This now proves that the curve X 10 8 4 82 thus obtained is a solution of the initial value problem dt 0