(b) If f satisfies the hypotheses of the contractive mapping principle and x1 is any point in M show that d(x1,x0) Solution Next we want to show, for any x0 ? X, that the recursively defined iterates xn = f(xn?1) for n ? 1 converge to a fixed point of f. How close is xn to xn+1? For any n ? 1, d(xn, xn+1) = d(f(xn?1), f(xn)) ? cd(xn?1, xn). Therefore d(xn, xn+1) ? cd(xn?1, xn) ? c^2d(xn?2, xn?1) ?.