Modulus Larger than Real Part

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

Then the modulus of $z$ is larger than the real part $\operatorname{Re} \left({z}\right)$ of $z$:
 * $\quad \left\vert{z}\right\vert \ge \left\vert{\operatorname{Re} \left({z}\right) }\right\vert$

Proof
By the definition of a complex number, we have:
 * $z = \operatorname{Re} \left({z}\right) + i \operatorname{Im} \left({z}\right)$

Then: