The Metric Structure and Sequences of Complex Numbers

If you’re going through hell, keep on going, Winston Churchill

Introduction

Definition. Let D ⊆ ℂ be a set of complex numbers. A complex-valued function f of a complex variable defined on D is a rule that assigns to each complex number z belonging to the set D a unique complex number w, $f: D \to \mathbb{C}$.

• Domain D: A subset of $\mathbb{C}$ on which the function is defined. Every $z \in D$ is called a point of the domain.
• Function $f: D \to \mathbb{C}$: A rule that assigns to each $z \in D$ a unique complex number w = f(z).
• Range (or image) f(D): The set of all actual outputs {f(z): z ∈ D}.
• Decomposition into real and imaginary parts: If z = x+ iy ∈ D, then f(z) is called the image of the point z under f. We often write f(z) = u(x ,y) + iv(x, y), where u, v: ℝ² → ℝ are the real and imaginary parts (real‑valued functions of two real variables).

The Metric Structure of ℂ

The complex plane ℂ carries a natural metric (distance function) inherited from its identification with ℝ².

Definition. For z₁, z₂ ∈ ℂ, the distance between them is: $d(z_1, z_2) = |z_1 - z_2| = \sqrt{(\text{Re}(z_1) - \text{Re}(z_2))^2 + (\text{Im}(z_1) - \text{Im}(z_2))^2}$

This metric satisfies the axioms:

1. Positivity: d(z₁, z₂) ≥ 0, with equality iff z₁ = z₂
2. Symmetry: d(z₁, z₂) = d(z₂, z₁)
3. Triangle inequality: d(z₁, z₃) ≤ d(z₁, z₂) + d(z₂, z₃)

Sequences of complex numbers

Definition. A sequence of complex numbers is a function a $\mathbb{N} \to \mathbb{C}$, mapping natural numbers to complex numbers.

• Notations: $(a_n)_{n=1}^{\infty}$, $(a_n)$, $\{a_n\}_{n=1}^{\infty}$ or simply $\{a_n\}$
• The value a(n) is written as aₙ and called the n-th term. It is a complex number $x_n + iy_n$ (Cartesian form) or $r_ne^{i\theta_n}$ (polar form).

Ways to define sequences

• Explicit formula. Direct computation from n, e.g., $a_n = \frac{1}{n} + \frac{i}{n^2}$. It converges to 0 as $n \to \infty$.
• Recursive definition. Terms depend on predecessors, e.g., $a_1 = 1, \; a_{n+1} = \frac{a_n + i}{2}$.
• Piecewise/Cases. There are different rules for different n, e.g., $a_n = \begin{cases} i^n & n \text{ even} \\ 1/n & n \text{ odd} \end{cases}$.
• Implicit. Defined by a property, e.g., $a_n$ = n-th digit of π.

Examples of Complex Sequences

• Geometric Sequence: $a_n = z^n \quad \text{for fixed } z \in \mathbb{C}$.
  (i) If |z| < 1: spirals inward toward the origin. Since $|z^n| = |z|^n \to 0 \text{ as } n \to \infty$, the sequence converges to 0 in modulus. The argument $arg(z^n) = n \cdot arg(z)$ causes rotational motion.
  (ii) If |z| = 1: rotates on unit circle. Since $|z^n| = |z|^n = 1$ for all n, so all terms lie on the unit circle. The argument evolves as $arg(z^n) = n \cdot arg(z)$.
  (iii) If |z| > 1: spirals outward from the origin to ∞.
• Arithmetic-Geometric: $a_n = \frac{n \cdot i^n}{2^n}$.
  $|a_n| = \frac{n}{2^n} \to 0 \text{ as } n \to \infty$ since exponential decay $2^n$ dominates linear growth n.
  Argument: $i^n = e^{\frac{in\pi}{2}}$ cycles through 1, i, −1, −i, so the sequence rotates while shrinking.
  Conclusion: $\lim_{n \to \infty}a_n = 0$ (converges to 0).
• Rational Function: $a_n = \frac{n^2 + in}{n^2 + 1} = \frac{1 + i/n}{1 + 1/n^2} \to 1 \text{ as } n \to \infty$
• Recursive (Mandelbrot-type): $z_0 = 0, \quad z_{n+1} = z_n^2 + c \quad \text{for fixed } c \in \mathbb{C}$
• Powers of Roots of Unity: $a_n = e^{2\pi i n/k} = \cos\frac{2\pi n}{k} + i\sin\frac{2\pi n}{k}$. This cycles through the k-th roots of unity. These points lie evenly spaced on the unit circle.
  Periodicity: $e^{2\pi i(n+k)/k}=e^{2\pi in/k}\cdot e^{2\pi i}=e^{2\pi in/k} \implies a_{n+k} = a_n$, so the sequence cycles through the kth roots of unity.
  If k = 1, $a_n=e^{2\pi in}=1, \forall n$, hence it is constant (converges). Otherwise, the sequence cycles through k distinct points on the unit circle. It never settles down to one value, so it does not converge. Periodicity kills convergence.
  Even though the sequence doesn’t converge for k>1, it does have limit points:
  (i) If k > 1, the set of limit points is exactly the set of all k-th roots of unity. (ii) If k = 1, the only limit point is 1.

Convergence of Sequences

Definition. A sequence $\{ a_n \}_{n=1}^∞$ is said to have a limit $L \in \mathbb{C}$ (or converges to a limit L) if $\forall \varepsilon > 0, \; \exists N \in \mathbb{N} \text{ such that } \forall n \geq N \Rightarrow |a_n - L| < \varepsilon$.

We write $\lim_{n \to \infty} a_n = L$, $a_n \to L$ or $a_n \xrightarrow{n \to \infty} L$.

A convergent sequence approaches a specific complex number L as n grows larger. a_n cluster around L as n grows larger.

Equivalently, for any disk B(L; ε) around L (no matter how small), all but finitely many terms of the sequence lie inside that disk. In other words, eventually all terms lie inside every disc B(L; ε)..

Formally, $\forall \varepsilon, \exist N \in \mathbb{N}$ such that $\forall n \ge N, a_n \in \mathbb{B}(a_n; \varepsilon)$. This is equivalent to the standard definition $\lim_{n \to \infty}a_n = L \iff \forall \epsilon > 0, \exist N \in ℕ \text{ such that } |a_n - L| < \epsilon, \forall n ≥ N$.

Imagine a target with center L. No matter how tiny you make the bullseye (radius ε), the “tail” of the sequence (all terms after a certain index N) will eventually land inside that bullseye and stay there forever.

Example. Consider the sequence $a_:=\frac{1}{n}+i(1−\frac{1}{n})$. As $n \to \infty, a_n \to 0+i(1−0)=i$. Thus, the limit L = i, and the sequence is convergent.

The following are all equivalent definitions of $a_n \to L$:

1. ε-N. $\forall \epsilon > 0, \exist N \in ℕ \text{ such that } |a_n - L| < \epsilon, \forall n ≥ N$.
2. Neighborhoods. Every neighborhood of L contains all but finitely many terms.
3. Distance. $d(a_n, L) \to 0$ as $n \to \infty$. The distance $d(a_n, L) = |a_n - L|$.
4. Component-wise Criterion. $a_n \to L \text{ in } \mathbb{C} \iff \begin{cases} x_n \to a \text{ in } \mathbb{R} \\ y_n \to b \text{ in } \mathbb{R} \end{cases}$.
5. Modulus & Argument . $|a_n| \to |L|$ and (if L ≠ 0) $arg(a_n) \to arg(L)$

The Component-wise Criterion. (Reduction to Real Sequences). A complex sequence converges if and only if its real and imaginary parts converge independently.

Let $a_n = x_n + i y_n$ where xₙ, yₙ ∈ ℝ, and let L = a + ib. Then, $a_n \to L \text{ in } \mathbb{C} \iff \begin{cases} x_n \to a \text{ in } \mathbb{R} \\ y_n \to b \text{ in } \mathbb{R} \end{cases}$.

This reduces complex convergence to two real convergence problems! It allows us to prove convergence in ℂ using standard tools from real calculus (like L’Hôpital’s rule on the parts).

Proof.

(⟹) Assume $a_n \to L$. For any ε > 0, there exists N such that n ≥ N implies |aₙ - L| < ε.

Since $|x_n - a| = |\text{Re}(a_n) - \text{Re}(L)| = |\text{Re}(a_n - L)| \leq |a_n - L| < \varepsilon$, we have $x_n \to a$.

Similarly, $|y_n - b| = |\text{Im}(a_n - L)| \leq |a_n - L| < \varepsilon$, so $y_n \to b$.

(⟸) Assume xₙ → a and yₙ → b. For any ε > 0:

• ∃N₁: n ≥ N₁ ⟹ $|xₙ - a| < \frac{\varepsilon}{\sqrt{2}}$
• ∃N₂: n ≥ N₂ ⟹ $|yₙ - b| < \frac{\varepsilon}{\sqrt{2}}$

Let N = max(N₁, N₂). For n ≥ N: $|a_n - L| = \sqrt{(x_n - a)^2 + (y_n - b)^2} < \sqrt{\frac{\varepsilon^2}{2} + \frac{\varepsilon^2}{2}} = \varepsilon$

Therefore $a_n \to L \blacksquare$

Example: $a_n = \frac{n}{n+1} + i \sin\left(\frac{\pi}{n}\right)$.
Real part: $x_n = \frac{n}{n+1} \to 1$.
Imaginary part: $y_n = \sin(\pi/n) \to \sin(0) = 0$.
Conclusion: $a_n \to 1 + 0i = 1$.

Uniqueness of Limits. If a sequence converges, its limit is unique.

Proof.

Suppose $aₙ \to L$ and $aₙ \to M$ with L ≠ M.

Consider $\varepsilon = \frac{|L-M|}{2}$.

Since aₙ → L: ∃N₁ such that n ≥ N₁ ⟹ |aₙ - L| < ε.

Since aₙ → M: ∃N₂ such that n ≥ N₂ ⟹ |aₙ - M| < ε.

For n ≥ max(N₁, N₂): $|L - M| \leq |L - a_n| + |a_n - M| < \varepsilon + \varepsilon = |L - M|$, $|L - M| < |L - M|, this is obviously a contradiction. ∎

Examples of Convergence

1. Spiral to the Origin. $a_n = \frac{1}{n}e^{in} = \frac{\cos n + i \sin n}{n}$
   |aₙ| = 1/n → 0. The terms spiral around the origin while approaching it.
   For any ε > 0, choose N > 1/ε. Then, for n ≥ N: $|a_n - 0| = \frac{1}{n} \leq \frac{1}{N} < \varepsilon$
2. Convergence Along Curve. $a_n = \frac{1}{n} + i\left(1 - \frac{1}{n}\right)$
   $a_n \to 1.$
   Proof using components: $Re(aₙ) = 1/n \to 0, Im(aₙ) = 1 - 1/n \to 1$. Therefore, $aₙ \to 0 + i(1) = i.$
3. Geometric Sequence: aₙ = zⁿ for fixed $z \in \mathbb{C}$.
   $\lim_{n \to \infty} z^n = \begin{cases} 0 & \text{if } |z| < 1 \\ 1 & \text{if } z = 1 \\ \text{diverges} & \text{if } |z| > 1 \\ \text{diverges} & \text{if } |z| = 1, z \neq 1 \end{cases}$
   Proof (case |z| < 1): Let r = |z| < 1. Then, |zⁿ| = rⁿ.
   For any ε > 0, we need rⁿ < ε, i.e., $n\cdot \log(r) \lt \log(\varepsilon) \implies[\text{Since log(r) < 0 (as 0 < r < 1)}] n > \frac{\log(\varepsilon)}{\log(r)}$.
   Choose $N = \frac{\log(\varepsilon)}{\log(r)}+1$. For n ≥ N: |zⁿ - 0| = rⁿ < ε. ∎

• Rational Function. $a_n = \frac{3n^2 + in}{n^2 + 2n + i}$
  $a_n = \frac{3 + i/n}{1 + 2/n + i/n^2} \to \frac{3 + 0}{1 + 0 + 0} = 3$
• Oscillating but Convergent. $a_n = \frac{(-1)^n}{n} + i\frac{1}{n}$. $a_n \to 0$
  $|a_n - 0| = \left|\frac{(-1)^n}{n} + \frac{i}{n}\right| = \frac{\sqrt{1 + 1}}{n} = \frac{\sqrt{2}}{n} \to 0$
  The oscillation in the real part doesn’t prevent convergence because the amplitude shrinks.

Divergence

Definition. A sequence $\{a_n\}_{n=1}^{\infty}$ converges to a limit $L \in \mathbb{C}$ if the terms eventually stay arbitrarily close to L. A sequence $\{a_n\}_{n=1}^{\infty}$ diverges if it does not converge to any $L \in \mathbb{C}$.

Taxonomy of Divergence

In complex analysis, “divergence” isn’t just one behavior. It helps to distinguish between two types:

1. Divergence to Infinity (Unbounded). The sequence marches away from the origin (or grows) without bound (as n increases). Equivalently, its terms eventually leave every bounded disk.
   $a_n \to \infty$ if $\forall M > 0, \exists N$ such that $\forall n \ge N \implies |a_n| > M$.
   Examples: $a_n = n + in$. $|a_n| = \sqrt{2}n \to \infty$.
   $aₙ = n + in²$. As $n \to \infty, ∣a_n∣ = \sqrt{n²+n⁴} \to \infty$.
   

   In the context of the Riemann Sphere ($\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$), this type is considered “convergence to the point at infinity,” while the second type remains truly divergent.

2. Bounded Divergence (Oscillation). The sequence stays within a finite region (it is bounded) but never settles on a single point (it has no limit).
   Example: $a_n = i^n = \{i, -1, -i, 1, \dots\}$
   It fits in the disk $B(0; 2)$, but has 4 different subsequential limits.

Examples of Divergence

• Polynomial Growth: $a_n = n + in^2$, $|a_n| = \sqrt{n^2 + n^4} = n\sqrt{1 + n^2} \geq n \to \infty$.
• Exponential Growth: $a_n = 2^n \cdot e^{in}$, $|a_n| = 2^n \to \infty$ (spiraling outward).
• Bounded Oscillation (no limit): $a_n = i^n = \begin{cases} 1 & n \equiv 0 \pmod{4} \\ i & n \equiv 1 \pmod{4} \\ -1 & n \equiv 2 \pmod{4} \\ -i & n \equiv 3 \pmod{4} \end{cases}$.
  This cycles through {1, i, -1, -i} forever—bounded but no limit.
• Real Oscillation: $a_n = (-1)^n$. Alternates between 1 and -1. Bounded (|aₙ| = 1) but divergent.
  Suppose $a_n \to L$ for some L ∈ ℂ. The subsequence $a_{2n} = 1 \to L$, so L = 1. However, $a_{2n+1} = -1 \to L$, so L = 1. Contradiction (Uniqueness of limits) ∎

Properties of Convergent Sequences

Theorem (Algebra of Limits). Suppose $aₙ \to L$ and $bₙ \to M$. Then:

1. Sum: $aₙ + bₙ \to L + M$
2. Difference: $aₙ - bₙ \to L - M$
3. Product: $aₙ \cdot bₙ \to L \cdot M$
4. Scalar multiple: $c \cdot aₙ \to c \cdot L$ for any c ∈ ℂ
5. Quotient: $aₙ/bₙ \to L/M$ (provided M ≠ 0 and bₙ ≠ 0 for all n)
6. Conjugate: $\overline{a_n} \to \overline{L}$ (continuity of conjugation).
7. Modulus: $|aₙ| \to |L|$ (continuity of modulus).
8. Real/Imaginary parts: $Re(aₙ) \to Re(L), Im(aₙ) \to Im(L)$

Theorem. Every convergent sequence is bounded.

An unbounded sequence cannot converge (to a finite limit).

Proof.

Suppose $a_n \to L$.

Taking ε = 1, there exists N such that n ≥ N implies |aₙ - L| < 1.

For n ≥ N: |aₙ| ≤ |aₙ - L| + |L| < 1 + |L|.

Let $M = max \{ |a_₁|, |a_₂|, \cdots, |a_{N-1}|, 1 + |L| \}.$

Then, |aₙ| ≤ M for all n. $\blacksquare$

The Squeeze/Sandwich Theorem. If |aₙ - L| ≤ bₙ for all n sufficiently large, and $b_n \to 0$, then $a_n \to L$.

Corollary. If |aₙ| ≤ cₙ where $c_n \to 0$, then $a_n \to 0.$

Examples:

For $a_n = \frac{\sin(n^2) + i\cos(n^3)}{n}$: $|a_n| \leq \frac{|\sin(n^2)| + |\cos(n^3)|}{n} \leq \frac{2}{n} \to 0$. Therefore, $aₙ \to 0$.

For $a_n = \frac{(-1)^n}{n} + i\frac{1}{n^2}, |a_n| \le \frac{1}{n}+\frac{1}{n^2} \to 0$, so $a_n \to 0$ despite oscillating real parts.

Polar Coordinates and Argument Convergence. For $a_n = r_n e^{i\theta_n} \to re^{i\theta}$:

• Modulus: $r_n \to r$ (since $|a_n| = r_n \to |L| = r$).
• Argument: $\theta_n \to \theta \text{ mod } 2 \pi$ if $r \ne 0$. If r = 0, $\theta_n$ may oscillate, e.g., $a_n = \frac{1}{n}e^{in}$.