Reverse Triangle Inequality/Real and Complex Fields/Proof 1
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Theorem
Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Then:
- $\cmod {x - y} \ge \size {\cmod x - \cmod y}$
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:
- $\map d {x, y} = \size {x - y}$
From the Reverse Triangle Inequality as applied to metric spaces:
- $(1): \quad \forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$
Let $z = 0$.
Then $(1)$ translates to:
- $\forall x, y, z \in X: \size {\size {x - 0} - \size {y - 0} } \le \size {x - y}$
Hence the result.
$\blacksquare$