I am looking for a reference of the following fact:
Let $\left(X,\left\Vert \cdot\right\Vert \right)$ be a Banach space. For every $\left(x^{n}\right)_{n\geq0}\subseteq X$ define $$\left\Vert \left(x^{n}\right)_{n\geq0}\right\Vert _{\Sigma}=\sum_{n=0}^{\infty}\left\Vert x^{n}\right\Vert $$
and let$$\mathcal{X}=\left\{ \left(x^{n}\right)_{n\geq0}\subseteq X:\,\left\Vert \left(x^{n}\right)_{n\geq0}\right\Vert _{\Sigma}<\infty\right\} .$$Then $\left(\mathcal{X},\left\Vert \cdot\right\Vert _{\Sigma}\right)$ is complete. Any reference would be appreciated. Thanks in advance.