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:

$\size {x - y} \ge \size x - \size y$

where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number.

Proof

From the Reverse Triangle Inequality:

$\cmod {x - y} \ge \cmod {\cmod x - \cmod y}$

By the definition of both absolute value and complex modulus:

$\cmod {\cmod x - \cmod y} \ge 0$

As:

$\cmod x - \cmod y = \pm \cmod {\cmod x - \cmod y}$

it follows that:

$\cmod {\cmod x - \cmod y} \ge \cmod x - \cmod y$

Hence the result.

$\blacksquare$