Prove ||u-v|| defines a metric space

Question

Given a normed linear space, I need to prove that $||U-V||$ defines a metric space. I can prove symmetry and positivity, but having trouble proving $||U-V||+||V-W||\geq ||U-W||$

I know :

$$||U-W||\leq ||U||+||W||\\||U-V||+|V-W|| \leq ||U||+||W||+2||V||$$

But, if I put them together, I cant seem to prove that $||U||+||W||\leq||U-V||+||V-W||$

Can anyone point me in the right direction?

Answer 1

$$||U-W||=||(U-V)+(V-W)|| \leq ||U-V||+||V-W||$$