Given a function $f(z)$ that is analytic over the whole complex plane and Im($f$) $\leq 0$, show that $f$ is constant

Question

By according to the Maximum Principle for Harmonic Functions, I could say that Im($f$)$= v$ is constant and, so, by using the Cauchy-Riemann equations, I prove that $f$ is constant over the whole complex plane.

Is it correct?

Could someone explain me another way of proving this?

Thanks in advance