1. The problem statement, all variables and given/known data xn+1 = xn + cosxn , n>=1 where x0 E [π/4 , 3π/4] = D. Show it converges, find rate of convergence. 2. Relevant equations contraction theorem 3. The attempt at a solution Setting a function f(x) = x+cosx we have f'(x) = 1 - sinx, f''(x)= -cosx. Now f' >= 0, so f is increasing. For x E D, the first derivative is less than one. Since f is increasing , f(π/4) = 1.4925 > π/4 and f(3π/4) = 1.6490 < 3π/4 , f is a mapping D -> D. So f is a contraction in D, thus the sequence xn+1 will converge to a fixed point x* Ε D. But doesn't the above hold for x0 only? Don't i have to prove that for all n, xn E D ?