Krull intersection theorem for Jacobson radical

Statement
Let $$R$$ be a Noetherian ring and $$I$$ an ideal contained inside the Jacobson radical of $$R$$. Then, we have:

$$\bigcap_{j=1}^\infty I^j = 0$$

In particular, when $$R$$ is a local ring, then the above holds for any proper ideal $$I$$.

Applying the Krull intersection theorem for modules
We apply the Krull intersection theorem for modules, which states that if $$R$$ is a Noetherian ring and $$M$$ is a finitely generated module over $$R$$, and $$I$$ is an ideal in $$R$$, we have:

$$I\left(\bigcap_{j=1}^\infty I^jM \right) = \bigcap_{j=1}^\infty I^jM$$

We apply it to the case $$M = R$$. We thus get:

$$I\left(\bigcap_{j=1}^\infty I^j \right) = \bigcap_{j=1}^\infty I^j$$

Applying Nakayama's lemma
Consider the ideal $$N = \bigcap_{j=1}^\infty I^j$$ as a $$R$$-module. Since $$IN = N$$, and $$I$$ is contained in the Jacobson radical of $$R$$, Nakayama's lemma tells us that $$N = 0$$. This is precisely what we want.