Reverse Triangle Inequality/Real and Complex Fields/Corollary 2

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: $3.2.5$