Reverse Triangle Inequality/Real and Complex Fields/Corollary/Proof 3

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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

Let $z_1$ and $z_2$ be represented by the points $A$ and $B$ respectively in the complex plane.

From Geometrical Interpretation of Complex Subtraction, we can construct the parallelogram $OACB$ where:

$OA$ and $OB$ represent $z_1$ and $z_2$ respectively
$BA$ represents $z_1 - z_2$.


Complex-Reverse-Triangle-Inequality-Corollary.png


But $OA$, $OB$ and $BA$ form the sides of a triangle.

The result then follows directly from Sum of Two Sides of Triangle Greater than Third Side.

$\blacksquare$


Sources