Proof of Limit Test

Statement of Limit Test

Let $\sum_{n=1}^\infty a_n$ converge. Then $a_n\to 0$ as $n \to \infty$.

Proof

For $k\in\mathbb{Z}^+$, let

$s_k = \sum_{n=1}^{k} a_n$

and also let

$\lim_{k\to\infty}s_k = s.$

By the definition of convergence, there exists $K\in\mathbb{Z}^+$ such that for $k>K$ we have

$|s_k - s| < \frac{\varepsilon}{2}$

for any $\frac{\varepsilon}{2} >0$. Now, noting that $a_{k+1} = s_{k+1} - s_k$ we have

$|a_{k+1}| = |s_{k+1} - s_{k} + s - s|$

and using the triangle inequality we deduce

$|a_{k+1}| \leq |s_{k+1} - s| + |s_{k} - s|.$

As both of these terms are strictly smaller than $\frac{\varepsilon}{2}$ we get

$|a_{k+1}| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2}$

$|a_{k+1}| < \varepsilon.$

As $\varepsilon$ is arbitrary, we deduce that $a_k\to 0$ as $k\to\infty.$