in Edward B. Saff, Arthur David Snider Fundamentals of complex analysis, with applications 2003, it has the following claim
Suppose we are given a function $f$ that is analytic and nonzero at each point of a simple closed cantour $C$ and is meromorphic inside $C$. Under these conditions it can be shown that $f$ has at most a finite number of poles inside $C$. The proof of this depends on two facts: first. the only singularities of $f$ are isolated singularities(poles), and, second, that every infinite sequence of points inside $C$ has a subsequence that converges to some point on or inside $C$. Hence if $f$ had an infinite numbers of poles inside $C$, some subsequence of them would converge to a point that must be a singularity, but not an isolated singularity of $f$.
I am struggling to understand why the bolded part is correct, why would a sequence of poles converge to an essential singularity?