$\def\Re{\mathop{\mathrm{Re}}}$Suppose $z \in \mathbb{C}$ satisfies $-\dfrac{1}{2} < \Re(z) < \dfrac{1}{2}$ and $|z| \geqslant 1$, and $c$ and $d$ are integers.
Why is it that $|cz + d| \leqslant 1$ and $|c| \geqslant 2$ are inconsistent with each other?
I have tried using algebraic manipulations but am unable to show this.
HINT
We can reduce to
$$|cz+d|^2=(cz+d)(c\bar z+d)=c^2|z|^2+cd(z+\bar z)+d^2=c^2|z|^2+2cdRe(z)+d^2 \ge c^2+2cdRe(z)+d^2 $$