Reverse Triangle Inequality/Real and Complex Fields/Proof 1

Theorem
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.

Then:
 * $\left\vert{x - y}\right\vert \ge \left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert$

where $\left\vert{x}\right\vert$ denotes either the absolute value of a real number or the complex modulus of a complex number.

Proof
Let $X$ denote either $\R$ or $\C$ as appropriate.

From Real Number Line is Metric Space and Complex Plane is Metric Space the distance function $d: X \times X \to \R$ can be defined as:
 * $d \left({x, y}\right) = \left\vert{x - y}\right\vert$

From the Reverse Triangle Inequality as applied to metric spaces:
 * $(1): \quad \forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$

Let $z = 0$.

Then $(1)$ translates to:
 * $\forall x, y, z \in X: \left|{\left\vert{x - 0}\right\vert - \left\vert{y - 0}\right\vert}\right| \le \left\vert{x - y}\right\vert$

Hence the result.