Prove $a$ approaches $0$, if $a<|b|$ and $b$ approaches $0$?

Question

Prove $a$ approaches $0$, if $a<|b|$ and $b$ approaches $0$?

It looks silly, but I can't write something down to prove it.

Answer 1

Let $a < \lvert b \rvert $. Assume that b approaches zero. This means that for any fixed $\epsilon$ we can make $b$ closer to zero than $\epsilon$ , which is the same thing as saying $\lvert b-0 \rvert\le\lvert\epsilon-0\rvert$, or just $\lvert b \rvert\le\lvert\epsilon\rvert$ . Thus if $a < \lvert b \rvert $, then by the transitive property $a\lt\epsilon$. Since this is any arbitary epsilon, $a$ must also approach zero