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A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy

Statement

Let $X$ be a set and $f_{n} : X \to C$ . Then $(f_{n})$ converges uniformly on $X$ if and only if it is uniformly Cauchy (Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary). This includes $X = \emptyset$ .

Facts & Assumptions

Given: A set $X$ and functions $f_{n} : X \to C$ .

[L1]

The complex plane is complete (The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts).

[L2]

For complex numbers, $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ and $∣ z ∣ = 0$ if and only if $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1L2

If $f_{n} \to f$ uniformly, then for $ε > 0$ choose $N$ with $∣ f_{n} (x) - f (x) ∣ < ε / 2$ for every $n \geq N$ and $x \in X$ ; for $m, n \geq N$ , [L2] gives $∣ f_{m} (x) - f_{n} (x) ∣ < ε$ , so $(f_{n})$ is uniformly Cauchy.

1.2L1choose

Conversely, suppose $(f_{n})$ is uniformly Cauchy. For each $x \in X$ , the sequence $(f_{n} (x))$ is Cauchy and hence has a limit $f (x) \in C$ by [L1]; this defines $f : X \to C$ , including the unique empty function when $X = \emptyset$ .

2.1step 1.2L2∎

Given $ε > 0$ , choose $N$ such that $∣ f_{m} (x) - f_{n} (x) ∣ < ε / 2$ for all $m, n \geq N$ and $x \in X$ . Fixing $n \geq N$ and passing $m \to \infty$ in the continuous modulus gives $∣ f (x) - f_{n} (x) ∣ \leq ε / 2 < ε$ for every $x$ , so $f_{n} \to f$ uniformly.

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)