The $\ell^p(\mathbb{C})$ spaces are complete

Statement

Let $p\in[1,+\infty]\subseteq$$\overline{\mathbb{R} }$ be given and consider $\ell^p(\mathbb{C})$ then we claim:

1. For $p\in\mathbb{R}_{\ge 1}$ that $$\ell^p(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \sum^\infty_{n\eq 1}\vert x_n\vert^p<+\infty\right\}\right.$$ is a complete metric space with respect to the metric induced by the norm: $$\Vert(x_n)_{n\in\mathbb{N} }\Vert_p:\eq\left(\sum^\infty_{n\eq 1}\vert x_n\vert^p\right)^\frac{1}{p}$$
2. For $p\eq+\infty$ that $$\ell^{\infty}(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)<+\infty \right\}\right.$$ is a complete metric space with respect to the metric induced by the norm: $$\Vert(x_n)_{n\in\mathbb{N} }\Vert_\infty:\eq\mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)$$

Proof of claims

Grade: A*

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