Absolute Value of Components of Complex Number no greater than Root 2 of Modulus

Theorem
Let $z = x + i y \in \C$ be a complex number.

Then:
 * $\size x + \size y \le \sqrt 2 \cmod z$

where:
 * $\size x$ and $\size y$ denote the absolute value of $x$ and $y$
 * $\cmod z$ denotes the complex modulus of $z$.

Proof
Let $z = x + i y \in \C$ be an arbitrary complex number.

the contrary:

But as $\size x$ and $\size y$ are both real this cannot happen.

Thus our initial assumption $\size x + \size y > \sqrt 2 \cmod z$ is false.

Hence the result.