Example for $\frac1{a_n}\to 0$ and $a_n$ not $\to\infty$

Question

Currently reading "How to think about analysis". It has a section where it says

do you see why theorem is true but converse is not? $$\text{Theorem: If }(a_n)\to\infty\text{ then }\left(\frac1{a_n}\right)\to 0$$

I was able to find couterexamples for converse of other theorems but couldn't find one for this. Any example for its converse?

Answer 1

For a sequence $(a_n)$ of nonzero real numbers we have $$ \frac{1}{a_n} \to 0 \iff \left| \frac{1}{a_n}\right | \to 0 \iff |a_n| \to \infty. $$ The latter implies $a_n \to \infty$ only if (all but finitely many of) the $a_n$ are positive.

Generally, we can have that $a_n \to -\infty$ (e.g. $a_n = -n$) or that $a_n$ has no limit at all (e.g. $a_n = (-1)^n n$).