Let R be a commutative ring, prove that each prime ideal of R contains the nilradical of R.

Proof so far.

Let I be a prime ideal of R, then if $\displaystyle xy \in I $, we would have either x or y in I.

Now, nilradical of R is an element $\displaystyle r \in R $ such that $\displaystyle r^n=0 \ \ \ n \geq 1 $.

How should approach this problem? I don't see how prime would lead to zero here. Thanks.