Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?
Prove that the sequence \(x_n = \sin (n^2)\) does not tend to zero for \(n\) that tends to infinity.