$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant

Let $f\colon\mathbb C \to \mathbb C$ be entire. Show that if
$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ for all $z \in \mathbb C$, then $f$ is constant on $\mathbb C$. How I can answer this by considering the distance between $f(z)$ and $i$.

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