Uniform convergence for a "sequence" of functions indexed by a real parameter

Question

For $-1 < r < 1$ we define $\psi_r \in \operatorname{Aut}(\mathbb{D})$ by $$\psi_r(z) = \frac{z+r}{1+rz}.$$

$\textbf{Lemma 14.3.}$ If $K$ is a compact subset of $\overline{\mathbb{D}} \setminus \{ -1 \}$, then $\psi_r \to 1$ uniformly on K as $r \to 1^-$.

I don't understand how uniform convergence for a "sequence" of functions indexed by the real parameter $r$ is defined.

Answer 1

It's quite analogous to the sequence case.

$\Psi_r(z)$ converges uniformly to $f(z)$ on $K$ as $r \to 1^-$ means for every $\epsilon > 0$ there is $\delta > 0$ such that when $1-\delta < r < 1$, $|\Psi_r(z) - f(z)| < \epsilon$ for all $z \in K$.