Nilradical of a Noetherian Ring is intersection of finitely many primes

Question

I am trying to prove that the nilradical of a Noetherian ring is the intersection of finitely many prime ideals but can not seem to do it.

I am trying to make a standard argument assuming, for contradiction, that the nilradical is not the intersection of finitely many prime ideals and deriving a contradiction but can't find one, any help is much appreciated.

Answer 1

Hint:

The nilradical is the intersection of all minimal prime ideals. So prove a noetherian ring has only a finite number of minimal primes.