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