Convergent Series of Natural Numbers

Theorem
Let $\left({a_n}\right)_{n \in \N}$ be a sequence of natural numbers.

Then the following are equivalent:

$(1): \quad \displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges

$(2): \quad \exists N \in \N: \forall n \ge N: a_n = 0$

That is, $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges iff only finitely many of the $a_n$ are non-zero.