Triangle Inequality for Complex Numbers/Corollary 3

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Corollary to Triangle Inequality for Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\cmod z$ denote the modulus of $z$.


Then:

$\cmod {z_1 - z_2} \le \cmod {z_1} + \cmod {z_2}$


Proof

Let $w = -z_2$.

Then:

\(\ds \cmod {z_1 + w}\) \(\le\) \(\ds \cmod {z_1} + \cmod w\) Triangle Inequality for Complex Numbers
\(\ds \leadsto \ \ \) \(\ds \cmod {z_1 + \paren {-z_2} }\) \(\le\) \(\ds \cmod {z_1} + \cmod {-z_2}\) Definition of $w$
\(\ds \leadsto \ \ \) \(\ds \cmod {z_1 - z_2}\) \(\le\) \(\ds \cmod {z_1} + \cmod {z_2}\) Absolute Value of Negative

$\blacksquare$


Sources

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