About the sequence x kezt we have the following 1 B x b for each te to b t b a MK 1 1 l b t b k20 k 1 c x kezt 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 C which is given by x 1 lim x 1 1 l b to b Thus x 1 lim x f s x s ds 565 8 60 ds b Finally the validity of this integral equation has the following two implications X t x s x s ds x 0 x 1 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 x 1 satisfying the integral equation x 1 x f s x s ds for all 1 t b t b 2 dx dt d 0 ff s x s ds f 1 X t This now proves that the curve X 1 8 1 82 thus obtained is a solution of the initial value problem