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

Routine facts are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity), and throughout, the results on real sequences cited below are read in the current definitions of sequences, convergence and Cauchy sequences, by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (iv); the results on conjugates and moduli cited below are read for the current complex numbers, conjugate and modulus, by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (ii), The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iii) and The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iv); and Cauchy sequences in a metric space are read in the current definition of metric spaces, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i).

Parts of differences. Everything below is reduced to real sequences through the real and imaginary parts, and the reduction rests on two facts about a difference $z-w$ of complex numbers. First, the parts are additive and commute with real scalars, by Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (ii); writing $z-w=z+(-1)w$, this gives

$$\operatorname{Re}(z-w)=\operatorname{Re}z-\operatorname{Re}w,\qquad\operatorname{Im}(z-w)=\operatorname{Im}z-\operatorname{Im}w .$$

Second, the modulus is controlled by the parts and controls them, by Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vi) applied to $z-w$:

$$|\operatorname{Re}(z-w)|\le|z-w|,\qquad|\operatorname{Im}(z-w)|\le|z-w|,\qquad|z-w|\le|\operatorname{Re}(z-w)|+|\operatorname{Im}(z-w)| .$$

Real and imaginary parts. We prove (ii). Suppose first that $z_{n}\to w$, and let $\varepsilon>0$. By Convergent Sequences of Complex Numbers (i) there is $N\in\mathbb{N}$ with $|z_{n}-w|<\varepsilon$ for every $n\ge N$, and then, for the same $n$, the first two inequalities above give

$$|\operatorname{Re}z_{n}-\operatorname{Re}w|=|\operatorname{Re}(z_{n}-w)|\le|z_{n}-w|<\varepsilon,$$

and likewise $|\operatorname{Im}z_{n}-\operatorname{Im}w|<\varepsilon$. So the same $N$ serves in Convergent Sequences of Real Numbers (i) for both $\operatorname{Re}z_{n}\to\operatorname{Re}w$ and $\operatorname{Im}z_{n}\to\operatorname{Im}w$. Conversely, suppose that $\operatorname{Re}z_{n}\to\operatorname{Re}w$ and $\operatorname{Im}z_{n}\to\operatorname{Im}w$, and let $\varepsilon>0$. Choose $N_{1}$ with $|\operatorname{Re}z_{n}-\operatorname{Re}w|<\varepsilon/2$ for $n\ge N_{1}$, then $N_{2}$ with $|\operatorname{Im}z_{n}-\operatorname{Im}w|<\varepsilon/2$ for $n\ge N_{2}$, and put $N=\max(N_{1},N_{2})$. For $n\ge N$ the third inequality above gives

$$|z_{n}-w|\le|\operatorname{Re}z_{n}-\operatorname{Re}w|+|\operatorname{Im}z_{n}-\operatorname{Im}w|<\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon,$$

so $z_{n}\to w$.

Uniqueness. For (i), let $z_{n}\to w$ and $z_{n}\to w'$. By (ii), $(\operatorname{Re}z_{n})$ converges to both $\operatorname{Re}w$ and $\operatorname{Re}w'$, so $\operatorname{Re}w=\operatorname{Re}w'$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (ii); in the same way $\operatorname{Im}w=\operatorname{Im}w'$; so $w=w'$, since a complex number is determined by its parts (Real and Imaginary Parts of a Complex Number (i)).

Arithmetic. For (iii), each statement is checked on the parts and transferred back by (ii). The constant sequence $(\lambda)$ has the constant sequences $(\operatorname{Re}\lambda)$ and $(\operatorname{Im}\lambda)$ as parts, which converge to $\operatorname{Re}\lambda$ and $\operatorname{Im}\lambda$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i); so $(\lambda)$ converges to $\lambda$. Now let $z_{n}\to w$ and $u_{n}\to v$, so that $\operatorname{Re}z_{n}\to\operatorname{Re}w$, $\operatorname{Im}z_{n}\to\operatorname{Im}w$, $\operatorname{Re}u_{n}\to\operatorname{Re}v$ and $\operatorname{Im}u_{n}\to\operatorname{Im}v$ by (ii). By Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (ii), $\operatorname{Re}(z_{n}+u_{n})=\operatorname{Re}z_{n}+\operatorname{Re}u_{n}$, which converges to $\operatorname{Re}w+\operatorname{Re}v=\operatorname{Re}(w+v)$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi); the imaginary parts behave in the same way, so $z_{n}+u_{n}\to w+v$ by (ii). The same argument with the formulas for parts of differences shows $z_{n}-u_{n}\to w-v$. For products, write $z=\operatorname{Re}z+(\operatorname{Im}z)i$ and $u=\operatorname{Re}u+(\operatorname{Im}u)i$; the product formula and the uniqueness of parts in The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii) give

$$\operatorname{Re}(zu)=\operatorname{Re}z\operatorname{Re}u-\operatorname{Im}z\operatorname{Im}u,\qquad\operatorname{Im}(zu)=\operatorname{Re}z\operatorname{Im}u+\operatorname{Im}z\operatorname{Re}u .$$

Applying this to $z_{n}u_{n}$ and using the rules for products, sums and differences in Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi), $\operatorname{Re}(z_{n}u_{n})\to\operatorname{Re}w\operatorname{Re}v-\operatorname{Im}w\operatorname{Im}v=\operatorname{Re}(wv)$ and $\operatorname{Im}(z_{n}u_{n})\to\operatorname{Im}(wv)$; hence $z_{n}u_{n}\to wv$ by (ii).

Real sequences. For (iv), let $y\in\mathbb{R}$. Then $\iota(y)=y+0i$, so $\operatorname{Re}\iota(y)=y$ and $\operatorname{Im}\iota(y)=0$ (Real and Imaginary Parts of a Complex Number (i)). Thus the parts of $(\iota(x_{n}))$ are $(x_{n})$ and the constant sequence $0$, and those of $\iota(x)$ are $x$ and $0$. By (ii), $\iota(x_{n})\to\iota(x)$ if and only if $x_{n}\to x$ and the constant sequence $0$ tends to $0$; the latter always holds, by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i), so $\iota(x_{n})\to\iota(x)$ if and only if $x_{n}\to x$.

Conjugates and moduli. For (v), let $z_{n}\to w$. For $z\in\mathbb{C}$, The Complex Conjugate of a Complex Number (i) gives $\overline{z}=\operatorname{Re}z+(-\operatorname{Im}z)i$, so, by the uniqueness in The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii),

$$\operatorname{Re}\overline{z}=\operatorname{Re}z,\qquad\operatorname{Im}\overline{z}=-\operatorname{Im}z .$$

Hence $\operatorname{Re}\overline{z_{n}}=\operatorname{Re}z_{n}\to\operatorname{Re}w=\operatorname{Re}\overline{w}$, and $\operatorname{Im}\overline{z_{n}}=-\operatorname{Im}z_{n}\to-\operatorname{Im}w=\operatorname{Im}\overline{w}$ by the rule for negatives in Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi); so $\overline{z_{n}}\to\overline{w}$ by (ii). For the moduli, the reverse triangle inequality of Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vii) gives

$$\big||z_{n}|-|w|\big|\le|z_{n}-w| .$$

Given $\varepsilon>0$, the $N$ that Convergent Sequences of Complex Numbers (i) provides for $z_{n}\to w$ therefore gives $\big||z_{n}|-|w|\big|<\varepsilon$ for $n\ge N$, which is $|z_{n}|\to|w|$ in the sense of Convergent Sequences of Real Numbers (i).

The Cauchy criterion. For (vi), suppose first that $(z_{n})$ is convergent, say $z_{n}\to w$ (Convergent Sequences of Complex Numbers (ii)), and let $\varepsilon>0$. Choose $N$ with $|z_{n}-w|<\varepsilon/2$ for $n\ge N$. For $m,n\ge N$, the triangle inequality of Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vii), together with $|w-z_{n}|=|-(z_{n}-w)|=|z_{n}-w|$ from Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii), gives

$$|z_{m}-z_{n}|\le|z_{m}-w|+|w-z_{n}|<\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon .$$

Conversely, suppose the condition of (vi) holds. Given $\varepsilon>0$, its $N$ also gives, by the inequalities for parts of differences,

$$|\operatorname{Re}z_{m}-\operatorname{Re}z_{n}|\le|z_{m}-z_{n}|<\varepsilon,\qquad|\operatorname{Im}z_{m}-\operatorname{Im}z_{n}|\le|z_{m}-z_{n}|<\varepsilon\qquad(m,n\ge N),$$

so $(\operatorname{Re}z_{n})$ and $(\operatorname{Im}z_{n})$ are Cauchy sequences in $\mathbb{R}$. By Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences (v) they converge, to some $a\in\mathbb{R}$ and $b\in\mathbb{R}$ respectively. Put $w=a+bi$; then $\operatorname{Re}w=a$ and $\operatorname{Im}w=b$ by Real and Imaginary Parts of a Complex Number (i), so $z_{n}\to w$ by (ii), and $(z_{n})$ is convergent.

The modulus metric. For (vii), recall that $d_{\mathbb{C}}(z,w)=|z-w|$ for all $z,w\in\mathbb{C}$, by Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (v). Hence, for every $\varepsilon>0$ and every $N\in\mathbb{N}$, the condition $d_{\mathbb{C}}(z_{n},w)<\varepsilon$ for all $n\ge N$ in Convergent Sequence in a Metric Space (i) is the condition $|z_{n}-w|<\varepsilon$ for all $n\ge N$ in Convergent Sequences of Complex Numbers (i), so the two notions of convergence to $w$ coincide. Likewise the condition $d_{\mathbb{C}}(z_{m},z_{n})<\varepsilon$ for all $m,n\ge N$ in Cauchy Sequence in a Metric Space (i) is the condition $|z_{m}-z_{n}|<\varepsilon$ for all $m,n\ge N$ in (vi), for the same $\varepsilon$ and $N$, so $(z_{n})$ satisfies the condition of (vi) exactly when it is a Cauchy sequence in $(\mathbb{C},d_{\mathbb{C}})$.