Let $a_n$ be a null sequence and let $b_n$ be a bounded sequence. Prove that $a_n \cdot b_n$ is a null sequence.
I tried using the product rule of sequences but cannot because $b_n$ is not necessarily convergent and may not have a limit. How do I go about answering this?
Best Answer
Let $\varepsilon>0$ be given.
Since the sequence $\{b_n\}$ is bounded, there is a positive number $M$ so that $|b_n| \leq M$ for all $n$.
Since $\underset{n \to \infty}\lim a_n = 0$, there is a positive integer $N$ so that $|a_n|<\varepsilon/M$ whenever $n \geq N$.
Hence $|a_n \cdot b_n|<\varepsilon$ whenever $n \geq N$.