Theorem 1: Let $(a_n)$ and $(b_n)$ be sequences. If $(a_n)$ converges to $0$ and $(b_n)$ is bounded then $(a_nb_n)$ converges to $0$.
• Proof: Let $\epsilon > 0$ be given.
• Since $(b_n)$ is bounded there exists an $M \in \mathbb{R}$, $M > 0$ such that $|b_n| \leq M$ for all $n \in \mathbb{N}$.
• Since $(a_n)$ converges to $0$ we have that for $\epsilon_1 = \frac{\epsilon}{M} > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:
• Then if $n \geq N$ we have that $(*)$ holds and so:
• Therefore $(a_nb_n)$ converges to $0$. $\blacksquare$