How to Prove the Triangle Inequality ||x|| - ||y|| ≤ ||x-y||?

[florenti]

Help me to show this inequality ?

Hi,

I have to show this inequality

| ||x||-||y|| | <= ||x-y||

I have tried to use the Couchy-Schwarz inequality but I didn't get anything.

Could anyone help me solving this.

Thanks a lot.

florent

 

[HallsofIvy]

The Cauchy-Schwarz inequality says that ||x+ y||\le ||x||+ ||y||.

If you let x= a-b and y= b, that becomes ||a-b+b||\le ||a- b||+ ||b|| or ||a||\le ||a- b|+ ||b|| which, subtracting ||b|| from both sides, gives ||a||- ||b||\le ||a- b||. You Should be able to use that.

 

[florenti]

Thanks a lot