I'm hoping someone can help me see why the Weyl Group of a root system is finite.

The definition I'm using:

Let (V,R) be a root system. Then the Weyl Group, W, is the group generated by reflections in the hyperplanes orthogonal to the roots.

I understand that W acts on R and R is finite. So if I could show that this action is in fact faithful (I believe it is?) then I would have that W is finite. I haven't been able to show this however.

Many thanks for any help!