<s>[INST] Proof of $ \sqrt{2}|z|\geq|Re(z)| + |Im(z)|$ I need to prove  $\sqrt{2}|z|\geq|Re(z)| + |Im(z)|$

This is what I tried to do :
$Let$ $z=x+iy$ , this gives us $|Re(z)| = |x|$ and $|Im(z)|=|y|$
Also , $|z| = \sqrt{x^2+y^2}$ 
Then we have $\sqrt{2}\sqrt{x^2+y^2} \geq x +y $
$\Rightarrow$ $\sqrt{2x^2+2y^2} \geq x +y$
$\Rightarrow$ $2x^2+2y^2 \geq (x +y)^2$
$\Rightarrow$ $2x^2+2y^2 \geq x^2 +2xy +y^2$
$\Rightarrow$ $x^2+y^2 \geq 2xy $

Is this correct so far ? How do I proceed from here ?
 [/INST] It should be $$\sqrt{2}\sqrt{x^2+y^2} \geq |x| +|y| $$
But if you write  $a= |x|$ and $b= |y|$ then $$\sqrt{2}\sqrt{a^2+b^2} \geq a +b $$
and the proof goes as you did.
</s>