About the sequence x keZ we have the following a x 1 EB x b

Last updated: 9/23/2023

About the sequence x keZ we have the following a x 1 EB x b for each te to b t b MKk X 00 017 te to b t bk20 k 1 c x keZ 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 Bx b n which is given by x1 lim x 1 tet b to b b Thus x t lim x ff s x s ds S 8 X 1 de x lim ff s x s ds x flim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function tx t satisfying the integral equation x 1 x f f s x s ds for all t 1 b t b Finally the validity of this integral equation has the following two implications 1 2 x s ds X t xo f s x s ds x 0 x dt 0 s x s ds 1 X 1 dt This now proves that the curve X 1 8 4 82 thus obtained is a solution of the initial value problem