Reverse Triangle Inequality/Real and Complex Fields/Corollary 2
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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 $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.
Let $z := -y$.
Then we have:
\(\ds \size {x - z}\) | \(\ge\) | \(\ds \size x - \size z\) | Reverse Triangle Inequality for Real and Complex Fields: Corollary $1$ | |||||||||||
\(\ds \size {x - \paren {-y} }\) | \(\ge\) | \(\ds \size x - \size {-y}\) | Definition of $z$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x + y}\) | \(\ge\) | \(\ds \size x - \size {-y}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x + y}\) | \(\ge\) | \(\ds \size x - \size y\) | as $\size y = \size {-y}$ |
Hence the result.
$\blacksquare$
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Triangle Inequalities: $3.2.5$