Cosine Function is Absolutely Convergent/Complex Case/Proof 2

Theorem

Let $z \in \C$ be a complex number.

Let $\cos z$ be the cosine of $z$.

Then:

$\cos z$ is absolutely convergent for all $z \in \C$.

Proof

Radius of Convergence of Power Series Expansion for Cosine Function shows that the radius of convergence of the complex cosine function is infinite.

Then Existence of Radius of Convergence of Complex Power Series shows that the complex cosine function is absolutely convergent.

$\blacksquare$