Problem
Assume that $\{x_n\}$ is bounded and $\lim\limits_{n \to \infty}y_n=0.$ Prove that $\lim\limits_{n \to \infty}x_ny_n=0.$
Proof
Since $\{x_n\}$ is bounded, then for all $n$ there exists $M>0$ such that $|x_n|<M,$ namely $-M<x_n<M$. Thus $-M\cdot |y_n|<x_ny_n<M\cdot|y_n|.$Since $\lim\limits_{n \to \infty}y_n=0,$ then $\lim\limits_{n \to \infty}|y_n|=0,$ which implies that $-M\cdot |y_n|$ and $M\cdot|y_n|$ have the same limit $0$ when $n \to \infty.$ By the squeeze theorem, we have $\lim\limits_{n \to \infty}x_ny_n=0.$