Convergence in the Complex Numbers by Real and Imaginary Parts, and Completeness in the Modulus Metric

In the setting of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers, let $d_{\mathbb{C}}$ be the distance $d_{\mathbb{C}}$ on $\mathbb{C}$, so that $(\mathbb{C},d_{\mathbb{C}})$ is a metric space by The Modulus Metric on the Complex Numbers (i), and let real and imaginary parts be as in Real and Imaginary Parts of a Complex Number (i). Let $(z_{n})$ be a sequence in $\mathbb{C}$.

(i) $(z_{n})$ is convergent in $(\mathbb{C},d_{\mathbb{C}})$ if and only if the sequences $(\operatorname{Re}z_{n})$ and $(\operatorname{Im}z_{n})$ in $\mathbb{R}$ are convergent.

(ii) In that case the limit of $(z_{n})$ in $(\mathbb{C},d_{\mathbb{C}})$ is

$$\Big(\lim_{n\to\infty}\operatorname{Re}z_{n}\Big)+\Big(\lim_{n\to\infty}\operatorname{Im}z_{n}\Big)i,$$

where $\lim_{n\to\infty}\operatorname{Re}z_{n}$ and $\lim_{n\to\infty}\operatorname{Im}z_{n}$ are the limits of these sequences in $\mathbb{R}$.

(iii) $(\mathbb{C},d_{\mathbb{C}})$ is complete.