I need to prove $\sqrt{2}|z|\geq|Re(z)| + |Im(z)|$

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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 $

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Is this correct so far ? How do I proceed from here ?