Terms in Convergent Series Converge to Zero

Theorem
Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Let the series $\displaystyle \sum_{n=1}^\infty a_n$ be convergent.

Then $a_n \to 0$ as $n \to \infty$.

Proof
Let $\displaystyle s = \sum_{n=1}^\infty a_n$.

Then $\displaystyle s_N = \sum_{n=1}^N a_n \to s$ as $N \to \infty$.

Also, $s_{N-1} \to s$ as $N \to \infty$. Thus:

Hence the result.