= Solution
For fixed $\tau=x+iy$, the set $\{|c\tau+d|:(c,d)\in\mathbb Z^2\setminus\{0\}\}$ has a positive minimum. Choose a primitive pair $(c,d)$ attaining it and complete it to a matrix $\gamma\in SL_2(\mathbb Z)$. Since
$$
\operatorname{Im}(\gamma\tau)=\frac{y}{|c\tau+d|^2},
$$
this point has maximal imaginary part in the orbit. Translate by a power of $T:\tau\mapsto\tau+1$ to arrange $|\operatorname{Re}\tau|\leq1/2$. If now $|\tau|<1$, applying $S:\tau\mapsto-1/\tau$ strictly increases the imaginary part, a contradiction. Thus $|\tau|\geq1$, proving that every orbit meets the <standard fundamental domain of the modular group> $\mathcal F$.
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