If $a_n$ is Cauchy then for $q\in\mathbb{Q}$, the sequence $(q$ $a_n)_n$ is Cauchy.

Question

If $a_n$ is Cauchy then for $q\in\mathbb{Q}$, the sequence $(q$ $a_n)_n$ is Cauchy.

Proof. Let $\varepsilon >0$. Assume $a_n$ is cauchy. Since $a_n$ is cauchy then there is a $N$ in $\mathbb{N}$ such that for all $n,m >N$ we have $\left| a_{n}-a_{n}\right| < \varepsilon$. So, xxx

Can you give a hint for proof?

Answer 1

Exploit the definition of $b_n=qa_n$: $$ |b_m-b_n|=|q||a_m-a_n|, $$ now note that $q$ does not depend on $n$, choose $\epsilon/|q|$ in the definition of Cauchy sequence (if $q$ is nonzero).