Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 1

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

Then:


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

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

Proof
From the Reverse Triangle Inequality:
 * $\left\vert{x - y}\right\vert \ge \left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert$

By the definition of both absolute value and complex modulus:
 * $\left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert \ge 0$

As:
 * $\left\vert{x}\right\vert - \left\vert{y}\right\vert = \pm \left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert$

it follows that:
 * $\left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert \ge \left\vert{x}\right\vert - \left\vert{y}\right\vert$

Hence the result.