Let $\mathbb{C}$ be the field of \reftext{def:complex-numbers-2026a}{complex numbers} and let $d_{\mathbb{C}}$ be the function assigning to each pair $z,w$ of complex numbers the \reftext{def:complex-modulus-2026a}{modulus} $|z-w|$, which is a \reftext{def:metric-space-2026a}{metric} on $\mathbb{C}$ by claim 9 of \ref{lem:complex-conjugate-modulus-properties-2026a}.

Then the metric space $(\mathbb{C},d_{\mathbb{C}})$ is \reftext{def:complete-metric-space-2026a}{complete}: every \reftext{def:cauchy-sequence-metric-space-2026a}{Cauchy sequence} in $(\mathbb{C},d_{\mathbb{C}})$ \reftext{def:convergent-sequence-metric-space-2026a}{converges} to a complex number.