Complex Cosine Function is Unbounded/Proof 2

Theorem

The complex cosine function is unbounded.

Proof

By Complex Cosine Function is Entire, we have that $\cos$ is an entire function.

Aiming for a contradiction, suppose that $\cos$ is a bounded function.

By Liouville's Theorem, we have that $\cos$ is a constant function.

However, by Cosine of Zero is One:

$\cos 0 = 1$

and by Cosine of Right Angle:

$\cos \dfrac \pi 2 = 0$

Therefore, $\cos$ is not a constant function.

This contradicts the statement that $\cos$ is a constant function.

We hence conclude, by Proof by Contradiction, that our assumption that $\cos$ is a bounded function is false.

Hence $\cos$ is unbounded.

$\blacksquare$