# Math Help - Cauchy Sequence Question

1. ## Cauchy Sequence Question

Prove Directly that, if $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ are Cauchy, so is $\{a_nb_n\}_{n=1}^{\infty}$.

I cannot use the same proof that shows that if a converges to A and b to B, then ab converges to AB.

2. ## Re: Cauchy Sequence Question

Originally Posted by Aryth
Prove Directly that, if $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ are Cauchy, so is $\{a_nb_n\}_{n=1}^{\infty}$.
Here is the basic trick
\begin{align*}|a_mb_m-a_nb_n| &\le |a_mb_m-a_nb_m|+|a_nb_m-a_nb_m|\\ &\le |a_m-a_n||b_m|+|a_n||b_m-b_n| \end{align*}