Reverse Triangle Inequality/Real and Complex Fields/Corollary 3

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Theorem

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

Then:

$\size {x + y} \ge \size {\size x - \size y}$

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:

\(\ds \size {x - z}\) \(\ge\) \(\ds \size {\size x - \size z}\) Reverse Triangle Inequality for Real and Complex Fields
\(\ds \size {x - \paren {-y} }\) \(\ge\) \(\ds \size {\size x - \size {-y} }\) Definition of $z$
\(\ds \leadsto \ \ \) \(\ds \size {x + y}\) \(\ge\) \(\ds \size {\size x - \size {-y} }\)
\(\ds \leadsto \ \ \) \(\ds \size {x + y}\) \(\ge\) \(\ds \size {\size x - \size y}\) as $\size y = \size {-y}$

Hence the result.

$\blacksquare$


Sources

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