1. ## Inequality

is it true that

$||z| - |w|| \leq |z + w|$??

if so what is the proof?

here is my workin so far.. please verify it thanks.

We know $|z+w| \leq |z|+|w|$
let c = z - w, so $|c+w| \leq |c|+|w|$

Now z = c + w,
so $|z| \leq |c|+|w|$
$|z| \leq |z-w| + |w|$
$|z-w| \geq |z| -|w|$

Now let d = - w,
so $|z - d| \geq |z| - |d|$
subbing in -w for d we get, $|z + w| \geq |z| - |d|$
subbing in |w| for |d| since they are equal, we get $|z + w| \geq ||z| - |w||$
(i added an extra modulus bracket outside the right hand side at the end of the equation).

End of proof.

2. $|x+y|\leq |x|+|y|$.
$|x| = |x-y+y|\leq |x-y|+|y|\implies |x|-|y|\leq |x-y|$
$|y|-|x|\leq |x-y|$
$||x|-|y||\leq |x-y|$.