Let $c_n$ be a sequence that 'keeps away from zero', that is, $|c_n|≥r \,\forall n$ for some $r>0$ .
Let $a_n$ be a sequence that converges to 0, but $|a_n|>0$ for all $n$.
I have to prove that the sequence $\frac{c_n}{a_n}$ is divergent.
I tried assuming it was convergent and going through a proof for the sequence being cauchy, but I could not get a sensible contradiction (is that an oxymoron?).
Is it really as simple as saying $|\frac{c_n}{a_n}| ≥ |\frac{r}{a_n}|$ and then giving some simple argument, as to why this should be divergent?