If $\{a_0, a_1, a_2, …\}$ has no limit point does this mean $|a_n| \rightarrow \infty?$

I am reading about meromorphic functions and it stated that a function that is meromorphic has a set $\{a_0, a_1, a_2, ...\}$ with poles at those points which have no limit point. My question is, if this set is infinite, is it true that $|a_n| \rightarrow \infty$ as $n \rightarrow \infty?$

• If $|a_n| \not\rightarrow \infty$ then the given sequence would contain an infinite bounded subsequence, which in turn would have at least one limit point. – Mirko Jun 10 '17 at 21:52
• The question doesn't make a lot of sense the way it is now... you seem to be assuming that the set of poles comes with/is an an enumeration. But it is not - it is a mere set without any natural kind of ordering, and not a sequence. – polynomial_donut Jun 10 '17 at 21:53

Thus the answer is yes: $\;|a_n|\xrightarrow[n\to\infty]{}\infty\;$