Mathematical Analysis Question: Cauchy sequences proof

Question

Let $(s_n)_n$ and $(a_n)_n$ be Cauchy sequences. Demonstrate that $(s_na_n)_n$ is a Cauchy sequence.

Answer 1

Have you seen a proof that the product of convergent sequences is convergent? This is nearly identical. You can write $$|s_n a_n - s_ma_m| = |s_n a_n - s_n a_m + s_n a_m - s_m a_m| \le |s_n| |a_n - a_m| + |a_m| |s_n - s_m|.$$

Use the fact that $\{s_n\}$ and $\{a_n\}$ are Cauchy, and the fact that Cauchy sequences are bounded.