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**davesface** Verify that $\displaystyle \sqrt[ ]{2} \left |z \right | \geq \left | Re(z) \right | + \left | Im(z) \right | $

Suggestion: reduce this inequality to $\displaystyle {( \left | x \right | - \left | y \right |)}^{2} \geq 0$

My steps:

$\displaystyle \sqrt[ ]{2} \left |z \right | \geq \left | x \right | + \left |y \right |$

$\displaystyle 2 {\left |z \right |}^{2} \geq {(\left | x \right | + \left |y \right |)}^{2}$

$\displaystyle 2 ({ x}^{2 } + { y}^{ 2}) \geq {\left | x \right | }^{ 2} + 2 \left | x \right | \left | y \right | + { \left | y \right |}^{2 } $

Where to go from here?