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
$\square$
