Complex Variables: Prove f is Constant

[CornMuffin]

Homework Statement

Let z be a complex variable

Suppose f is an entire function and $$Re(f(z))\leq c$$ for all z

Show that f is constant.
(Hint: Consider exp(f(z))

Homework Equations

possibly this: $$e^z=e^x(cos(y)+isin(y))$$ where $$z=x+iy$$

The Attempt at a Solution

I had no idea how I would show this, so I just started off trying a few things:
I first started off working with the hint to consider exp(f(z)), where exp((f(z))=e^f(z)
I set g(z) equal to exp((f(z)) and because f(z) is entire, g(z) would also have to be entire
I first found a formula for the derivative of g(z) but that got me nowhere

I also tried working off the fact that $$Re(g(z))\leq e^ccos(Im(f(z)))$$
but that got me nowhere as well...

I have been thinking about this problem for so long now, and I couldn't think of a way to show that f is constant

 

[Dick]

Do you know Liouville's theorem? |exp(f(z))|<=exp(c).