# Triangle Inequality/Complex Numbers/Corollary 1

## Theorem

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

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

Then:

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

## Proof

Let $z_3 := z_1 + z_2$.

Then:

 $\ds \cmod {z_3} + \cmod {\paren {-z_2} }$ $\ge$ $\ds \cmod {z_3 + \paren {-z_2} }$ Triangle Inequality for Complex Numbers $\ds \leadsto \ \$ $\ds \cmod {z_3} + \cmod {z_2}$ $\ge$ $\ds \cmod {z_3 - z_2}$ Complex Modulus of Additive Inverse $\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 \ \$ $\ds \cmod {z_1 + z_2}$ $\ge$ $\ds \cmod {z_1} - \cmod {z_2}$

$\blacksquare$