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:
 * $\left|{x}\right| + \left|{y}\right| \le \sqrt 2 \left|{z}\right|$

where:
 * $\left|{x}\right|$ and $\left|{y}\right|$ denote the absolute value of $x$ and $y$
 * $\left|{z}\right|$ denotes the complex modulus of $z$.

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

Aiming for a contradiction, suppose the contrary:

But as $\left\vert{x}\right\vert$ and $\left\vert{y}\right\vert$ are both real this cannot happen.

Thus our initial assumption $\left\vert{x}\right\vert + \left\vert{y}\right\vert > \sqrt 2 \left\vert{z}\right\vert$ is false.

Hence the result.