Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 2

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Example of Use of Triangle Inequality for Complex Numbers

For all $z_1, z_2, z_3 \in \C$:

$\cmod {z_1 + z_2 + z_3} \le \cmod {z_1} + \cmod {z_2} + \cmod {z_3}$


Proof

\(\ds \cmod {z_1 + z_2 + z_3}\) \(=\) \(\ds \cmod {z_1 + \paren {z_2 + z_3} }\)
\(\ds \) \(\le\) \(\ds \cmod {z_1} + \cmod {z_2 + z_3}\) Triangle Inequality for Complex Numbers
\(\ds \) \(\le\) \(\ds \cmod {z_1} + \cmod {z_2} + \cmod {z_3}\) Triangle Inequality for Complex Numbers

$\blacksquare$


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