Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2

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
By the Triangle Inequality:
 * $\left\vert{x + y}\right\vert - \left\vert{y}\right\vert \le \left\vert{x}\right\vert$

Let $z = x + y$.

Then $x = z - y$ and so:


 * $\left\vert{z}\right\vert - \left\vert{y}\right\vert \le \left\vert{z - y}\right\vert$

Renaming variables as appropriate gives:
 * $\left\vert{x - y}\right\vert \ge \left\vert{x}\right\vert - \left\vert{y}\right\vert$