Corollary 2

Let A be a commutative Noetherian ring and a, b two ideals of A. There exists an integer h > 0 such that $a^h \cap b \subset ab$.

There exists n such that $a^{n+1} \cap b = a(a^n \cap b) \subset ab$ by Corollary 1 applied to $E = A, F = b$.