Triangle Inequality for Complex Numbers/Corollary 2

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Theorem

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

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


Then:

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


Proof

\(\ds \cmod {z_1 + z_2}\) \(\ge\) \(\ds \cmod {z_1} - \cmod {z_2}\) Triangle Inequality for Complex Numbers: Corollary $1$
\(\ds \cmod {z_1 + z_2}\) \(\ge\) \(\ds \cmod {z_2} - \cmod {z_1}\) Triangle Inequality for Complex Numbers: Corollary $1$
\(\ds \) \(=\) \(\ds -\paren {\cmod {z_1} - \cmod {z_2} }\)
\(\ds \leadsto \ \ \) \(\ds \cmod {z_1 + z_2}\) \(\ge\) \(\ds \cmod {\cmod {z_1} - \cmod {z_2} }\) Negative of Absolute Value: Corollary $2$

$\blacksquare$


Sources

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