Show that if ${a_n}$ and ${b_n}$ are sequences for which $\displaystyle\lim_{n\to\infty}a_n=0$ and ${b_n}$ is bounded, then $\displaystyle\lim_{n\to\infty}a_nb_n=0$
Sorry I'm on my phone and I'm not sure how to do all the correct notation. My idea was to use the theorem that says $\displaystyle\lim_{n\to\infty}a_nb_n=\left(\displaystyle\lim_{n\to\infty}a_n\right)\left(\displaystyle\lim_{n\to\infty}b_n\right)=0\cdot\left(\displaystyle\lim_{n\to\infty}b_n\right)$. But I have no idea how to prove this theorem and I am not sure how to incorporate the fact that b I bounded.
hint: $0\leq |a_nb_n| \leq K|a_n|, n \geq N_0$.