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## Homework Statement

Let [tex]f=u+iv[/tex] be an entire function. Prove that if [tex]u[/tex] is bounded, then [tex]f[/tex] is constant.

## Homework Equations

Liouville's Theorem states that the only bounded entire functions are the constant functions on [tex]\mathbb{C}[/tex]

## The Attempt at a Solution

I know that if [tex]u[/tex] is bounded, then the real part of the function is bounded, obviously. I need a function that is bounded for both the real and imaginary parts when just the real part is bounded and I am not sure how to find that...