1. Complex variable inequality

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?

2. Originally Posted by 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?
Remember that x and y are real numbers, so that $\displaystyle x^2 = |x|^2$ and $\displaystyle y^2 = |y|^2$. If you then take everything over to the left side of that last inequality, it becomes $\displaystyle \bigl(|x|-|y|\bigr)^2\geqslant0$, which is obviously true.

3. Wow, that was blindingly obvious. Thanks for the tip.