ThmDex – An index of mathematical definitions, results, and conjectures.

Let $x : \mathbb{N} \to \mathbb{R}$ ne a D4685: Real sequence.

Suppose that $\sum_{n = 0}^{\infty} x_n = a$ for some $a \in \mathbb{R}$ and fix $\varepsilon > 0$. Since the series $N \mapsto \sum_{n = 0}^N x_n$ is now a convergent sequence in $\mathbb{R}$, then result R244: Convergent sequence is Cauchy says that it is a Cauchy sequence. That is, there exists a threshold $M \geq 1$ such that \begin{equation} |x_N| = \left| \sum_{n = 0}^N x_n - \sum_{n = 0}^{N - 1} x_n \right| < \varepsilon \end{equation} for every $N \geq M$. Since $\varepsilon > 0$ was arbitrary, $x$ converges to $0$ due to R1089: Characterisation of convergent sequences in metric space.

The second claim is a consequence of the results

(i)	R2955: Reciprocal positive integer sequence converges to zero
(ii)	R2954: Standard real harmonic series diverges

$\square$

(1)	\begin{equation} \sum_{n = 0}^{\infty} x_n \in \mathbb{R} \quad \implies \quad \lim_{n \to \infty} x_n = 0 \end{equation}
(2)	\begin{equation} \lim_{n \to \infty} \frac{1}{n} = 0, \quad \text{ but } \quad \sum_{n = 0}^{\infty} \frac{1}{n} = \infty \end{equation}