Let S be in An(k) be a subset. Show that I(S) is radical.

I understand the forward direction, but am having trouble with the other way: rad(I(S)) is contained in I(S)

I started with:
Let f be in rad(I(S))
Then, fM is in I(S) for some M a natural number
I don't know whether I should try and do something with varieties from here, or not.

Thanks for any help you can offer!