Terms in Convergent Series Converge to Zero

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

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

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

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

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

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

$$ $$ $$

Hence the result.