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

florenti · Nov 8, 2008

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 · Nov 8, 2008

The Cauchy-Schwarz inequality says that [itex]||x+ y||\le ||x||+ ||y||[/itex].

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

florenti · Nov 8, 2008

Thanks a lot

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

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