Proof of Uniqueness of the Complex Derivative

Routine facts about real and complex numbers are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity).

A ball inside $U$. Since $U$ is open, there is a real number $r>0$ such that the open ball $B_{d_{\mathbb{C}}}(z,r)$ is contained in $U$ (Open Subset of a Metric Space (i), Open Ball in a Metric Space (i), with $d_{\mathbb{C}}$ as in Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (v)); that is, every $w\in\mathbb{C}$ with $|w-z|<r$ lies in $U$.

A point close to $z$. Let $\varepsilon>0$; we choose in this order. First take $\delta_{L}>0$ from the hypothesis for $N=L$ and $\delta_{M}>0$ from the hypothesis for $N=M$, both for this $\varepsilon$, and let $\delta=\min(\delta_{L},\delta_{M})$. Then put

$$t=\tfrac{1}{2}\min(r,\delta),\qquad w=z+t .$$

The modulus of the real number $t$ is its absolute value (Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (v)), so $0<|w-z|=t<\min(r,\delta)$. In particular $w\in U$ by the choice of $r$, and $0<|w-z|<\delta_{L}$ and $0<|w-z|<\delta_{M}$.

Comparing $L$ and $M$. Let

$$Q=\frac{f(w)-f(z)}{w-z}.$$

By the choice of $\delta_{L}$ and $\delta_{M}$ we have $|Q-L|<\varepsilon$ and $|Q-M|<\varepsilon$, and the triangle inequality (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vii)) gives

$$|L-M|\le|Q-L|+|Q-M|<2\varepsilon .$$

Thus $|L-M|<2\varepsilon$ for every real $\varepsilon>0$.

Conclusion. Suppose that $L\neq M$. Then $|L-M|>0$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii)), and the choice $\varepsilon=|L-M|/2$ in the previous paragraph gives $|L-M|<|L-M|$, a contradiction. Hence $L=M$.