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About the sequence x keZ we have the following a x 1 B x b...

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Algebra

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About the sequence x keZ we have the following a x 1 B x b for each 1 to b to b MK b x 1 x1 1 to b t b k 20 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 B x b which is given by x t lim x 1 11 b t b Thus X 1 Jim A 506 8 5 de Finally the validity of this integral equation has the following two implications X 1 xo f s x s ds x 0 x 1 x lim ff s x s ds x flim f s x s ds x f f s lim x s ds xo ff s x s ds Thus the function tx 1 satisfying the integral equation x 1 x f s x s ds for all t 16 b 1 b 2 dt 0 ff s x s ds f t X t This now proves that the curve X 1 6 82 thus obtained is a solution of the initial value problem

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