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

Theorem

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

Then:

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

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

Proof

By the Triangle Inequality:

$\cmod {x + y} - \cmod y \le \cmod x$

Let $z = x + y$.

Then $x = z - y$ and so:

$\cmod z - \cmod y \le \cmod {z - y}$

Renaming variables as appropriate gives:

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

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.18$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $7 \ \text{(c)}$