Nilradical is smallest radical ideal

Statement
The nilradical of a commutative unital ring (the set of nilpotent elements) is a radical ideal, and is contained in every radical ideal.

Proof
The proof involves three observations:


 * 1) The nilradical is an ideal. This is the hardest to prove, and uses commutativity.
 * 2) The nilradical is a radical ideal. This is because if a power of an element is nilpotent, then so is that element
 * 3) Any radical ideal contains the nilradical. This is because any radical ideal contains $$0$$, and hence must contain any nilpotent element since any nilpotent element has a power inside the ideal