Analytic functions and constants: Proving $f(z)-g(z)$ is constant

Question

Let $f(z)$ and $g(z)$ be analytic on some domain. Show that if $\Re(f(z)) = \Re(g(z))$ then $f(z)-g(z)$ is constant.

I haven't a clue on how to start. What is being asked of me & What am I looking to do?

Answer 1

If you write the difference $f(z)-g(z) = u(z) + iv(z)$, the condition that $\mathrm{Re}(f(z)) = \mathrm{Re}(g(z))$ implies that $u(z)=0$.

Hint: What does Cauchy-Riemann tells you, knowing that $f(z)-g(z)=iv(z)$ is holomorphic?