Thread: Inequality help

Hey there. Trying to understand this given inequality because I am supposed to prove it. I just want a starting point.

||x-y|| ||z|| =< ||x-z|| ||y|| + ||z-y|| ||x||

so what i did so far is basically expand the norms on the left and right but I'm not really sure what to do from here. How can I show this is true? Any tips?

hint: square both sides to simplify the algebra...