Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in\mathbb{C}$. Prove $f(z)=$constant 
Possible Duplicate:
Liouville's theorem problem 

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and suppose $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in\mathbb{C}$. How would you prove the function is constant?
I am approaching it by attempting to show it is bounded then by applying Liouville's Theorem. But have not made any notable results yet, any help would be greatly appreciated!

Since this seems like homework (if it is, you should use the homework tag), I will only give a hint.
Think about what the image domain of the function will look like. Can you postcompose $f$ with a simple holomorphic function on this domain (e.g. a Möbius transformation) such that the new function $g$ you obtain is bounded?
