# Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I tried playing with the properties but got nowhere. Now I'm beginning to have doubts over the fact that it is true at all. Is it? If so, could you provide a proof (which, by the way, I haven't found googling)?

Norms satisfy $$\lVert x+y\rVert \leqslant \lVert x\rVert +\lVert y\rVert$$
Then let $x=x'-y',y=y'-z'$. What we're saying is every norm induces a metric. But note the converse is not true.