Is it true that ||z| - |w|| \leq |z + w| ?

[smoothman]

is it true that ||z| - |w|| \leq |z + w| ??

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.

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

 

[leon1127]

|z| =|z+w-w|<= |z+w|+|w|

go from here