# Why there is a minimal element of this set

I'm trying to understand this proof:

I know intuitively, but Why formally there is such a minimal element?

I need help

Thanks

• That's from Fulton's book on Algebraic Curves no? The answer to your question is exactly above the theorem you are sharing with us, i.e. between the Lemma and Theorem 2 on page 8. Feb 15 '14 at 13:14

Because we are supposing this set is non-empty and $\mathbb A^n$ is noetherian, the result follows easily.