12.6.2 Higher mathematics

Problem: Prove that the sequence \( n \sin (n) \) has a bounded subsequence. Hint: \( \quad \) use the assertion \( \forall \alpha \in \mathbb{R}, \alpha>0 \), \( \forall M \in \mathbb{N} \exists p, q \in \mathbb{N}, \quad q>M, \quad|\alpha-p / q| \leq 1 / q^{2} \). In other words, any real number is approximated up to \( q^{-2} \) by an infinite set of rational numbers of the form \( p / q \).