IMC 2010 · Problem 1

(a) A sequence $x_1, x_2, \dots$ of real numbers satisfies

$$x_{n+1} = x_n \cos x_n \quad \text{for all } n \ge 1.$$

Does it follow that this sequence converges for all initial values $x_1$?

(b) A sequence $y_1, y_2, \dots$ of real numbers satisfies

$$y_{n+1} = y_n \sin y_n \quad \text{for all } n \ge 1.$$

Does it follow that this sequence converges for all initial values $y_1$?