Define the following fractional linear transformations (acting on elements of $\mathbb C$):
- $T_2:\tau \mapsto \tau + 2$
- $S: \tau \mapsto -1/\tau$
Let $G$ be the group of transformations generated by $T_2$ and $S$.
In my complex analysis textbook, there is a proof that for each point $\tau$ in the upper half-plane, there exists a mapping $g \in G$ such that $g(\tau) \in \mathcal F = \{\tau \in \mathbb C: |\Re(\tau)| \leq 1 \text{ and } \Im(\tau) \geq 0 \text{ and } |\tau| \geq 1 \}$.
Suppose we are given some point $\tau$. The proof begins by choosing $g \in G$ such that $\Im(g(\tau))$ is maximal. My question is, how do we know that there exists such a mapping $g$? (What if, for every $g \in G$, we can find a $g_1 \in G$ such that $\Im(g(\tau)) < \Im(g_1(\tau))$?