Terms in Convergent Series Converge to Zero

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Theorem

Let $\left \langle {a_n} \right \rangle$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$.

Suppose that the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$.

Then:

$\displaystyle \lim_{n \mathop \to \infty} a_n = 0$

Proof

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

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

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

Thus:

 $\displaystyle a_N$ $=$ $\displaystyle \paren {a_1 + a_2 + \cdots + a_{N - 1} + a_N} - \paren {a_1 + a_2 + \cdots + a_{N - 1} }$ $\displaystyle$ $=$ $\displaystyle s_N - s_{N - 1}$ $\displaystyle$ $\to$ $\displaystyle s - s = 0 \text{ as } N \to \infty$

Hence the result.

$\blacksquare$