Triangle Inequality for Complex Numbers/Corollary 1

Theorem

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

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

Then:

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

Let $z_3 := z_1 + z_2$.

Then:

$\ds \cmod {z_3} + \cmod {\paren {-z_2} }$

$\ge$

$\ds \cmod {z_3 + \paren {-z_2} }$

$\ds \leadsto \ \ $

$\ds \cmod {z_3} + \cmod {z_2}$

$\ge$

$\ds \cmod {z_3 - z_2}$

$\ds \leadsto \ \ $

$\ds \cmod {z_1 + z_2} + \cmod {z_2}$

$\ge$

$\ds \cmod {z_1}$

substituting $z_3 = z_1 + z_2$

$\ds \leadsto \ \ $

$\ds \cmod {z_1 + z_2}$

$\ge$

$\ds \cmod {z_1} - \cmod {z_2}$

$\blacksquare$