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. For full proof, refer: Nilradical is an ideal
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