To prove Theorem 5.20, part (i), show that the hypotheses imply that there exists a constant K

Question:

To prove Theorem 5.20, part (i), show that the hypotheses imply that there exists a constant K > 0 such that
|ui − vi| ≤ K|u0 − v0|, for each 1 ≤ i ≤ N,
whenever {ui}Ni =1 and {vi}Ni=1 satisfy the difference equation wi+1 = wi + h((ti ,wi , h).