Complex Cosine Function is Unbounded

Theorem
The complex cosine function is unbounded.

Proof
Let $K \in \R_{>0}$ be an arbitrary real number.

Let $p = \ln {2 K}$.

Let $z = i p$, where $i$ denotes the imaginary unit.

Then:

Thus for any $K$ we can find $z \in \C$ such that $\cmod {\cos z} > K$.

Hence the result by definition of unbounded.