Uniqueness of the Complex Derivative

In the setting of Complex Analysis: Standing Notation, let $U$ be an open subset of the metric space $(\mathbb{C},d_{\mathbb{C}})$, let $f:U\to\mathbb{C}$ be a function, and let $z\in U$.

(i) Uniqueness. Let $L,M\in\mathbb{C}$, and suppose that each of $N=L$ and $N=M$ has the following property: for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $w\in U$ with $0<|w-z|<\delta$ satisfies

$$\Bigl|\frac{f(w)-f(z)}{w-z}-N\Bigr|<\varepsilon ,$$

the quotient being defined because $w\neq z$. Then $L=M$.