Interesting algorithm for computing sqrt(2)?

Let x = 1 and for n ≥ 1 define x(n+1)= (x(n) + 2)/(x(n) + 1). Show that x converges, with limit sqrt(2). (Hint: Show that |x(n) - sqrt(2)| is monotone decreasing, and apply the Monotone Convergence Theorem to yn = |x(n) - sqrt(2)|)