is it true that

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

if so what is the proof?

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

We know $\displaystyle |z+w| \leq |z|+|w| $

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

Now z = c + w,

so $\displaystyle |z| \leq |c|+|w| $

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

$\displaystyle |z-w| \geq |z| -|w| $

Now let d = - w,

so $\displaystyle |z - d| \geq |z| - |d| $

subbing in -w for d we get, $\displaystyle |z + w| \geq |z| - |d| $

subbing in |w| for |d| since they are equal, we get $\displaystyle |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.

Is this correct please? please guide me if i am wrong?