The Complex Numbers are Complete in the Modulus Metric

Let $\mathbb{C}$ be the field of complex numbers and let $d_{\mathbb{C}}$ be the function assigning to each pair $z,w$ of complex numbers the modulus $|z-w|$, which is a metric on $\mathbb{C}$ by claim 9 of Properties of Complex Conjugation and Modulus.

Then the metric space $(\mathbb{C},d_{\mathbb{C}})$ is complete: every Cauchy sequence in $(\mathbb{C},d_{\mathbb{C}})$ converges to a complex number.