Let be $\{x_n \}$ and $\{y_n\}$ two sequences. If $\lim x_n=0$ and $\{y_n\}$ is a bounded sequence, then $\lim x_n y_n =0$

We have there exists $c \in \mathbb{R}$ such that $|y_n| < c$ for all $n \in \mathbb{N}$. Given $\varepsilon > 0$, then we can obtain $n_0 \in \mathbb{N}$ such that $n > n_0 \Longrightarrow |x_n| < \frac{\varepsilon}{c}$. Therefore, $n > n_0 \Longrightarrow |x_n \dot y_n | = |x_n| |y_n| < \frac{\varepsilon}{c} c = \varepsilon$. This implies that $x_n y_n \rightarrow 0$