Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric

In the setting of Complex Analysis: Standing Notation, let convergence of sequences in $\mathbb{C}$ be as in Convergent Sequences of Complex Numbers (i) and Convergent Sequences of Complex Numbers (ii). Let $(z_{n})$ and $(u_{n})$ be sequences in $\mathbb{C}$, let $(x_{n})$ be a sequence in $\mathbb{R}$, and let $w,v,\lambda\in\mathbb{C}$ and $x\in\mathbb{R}$.

(i) Uniqueness. If $z_{n}\to w$ and $z_{n}\to w'$ with $w'\in\mathbb{C}$, then $w=w'$.

(ii) Real and imaginary parts. $z_{n}\to w$ if and only if $\operatorname{Re}z_{n}\to\operatorname{Re}w$ and $\operatorname{Im}z_{n}\to\operatorname{Im}w$ in $\mathbb{R}$.

(iii) Arithmetic. The constant sequence $(\lambda)_{n\in\mathbb{N}}$ converges to $\lambda$. If $z_{n}\to w$ and $u_{n}\to v$, then

$$z_{n}+u_{n}\to w+v,\qquad z_{n}-u_{n}\to w-v,\qquad z_{n}u_{n}\to wv.$$

(iv) Real sequences. $\iota(x_{n})\to\iota(x)$ if and only if $x_{n}\to x$ in $\mathbb{R}$.

(v) Conjugates and moduli. If $z_{n}\to w$, then $\overline{z_{n}}\to\overline{w}$ and $|z_{n}|\to|w|$.

(vi) The Cauchy criterion. $(z_{n})$ is convergent if and only if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|z_{m}-z_{n}|<\varepsilon$ for all $m,n\in\mathbb{N}$ with $m,n\ge N$.

(vii) The modulus metric. $z_{n}\to w$ if and only if $(z_{n})$ converges to $w$ in $(\mathbb{C},d_{\mathbb{C}})$, and the condition on $(z_{n})$ in (vi) holds if and only if $(z_{n})$ is a Cauchy sequence in $(\mathbb{C},d_{\mathbb{C}})$.