Let $R$ be a complete local ring with respect to maximal ideal $I$. Then, intersection of $I^n$ is zero.

Question

Let $R$ be a complete local ring with respect to maximal ideal I.

I would like to prove intersection of $I^n$, $n\ge1$ is zero.

My attempt : $R$ is complete with respect to $I$, so $R$ is Hausdorff space. This is equivalent to $\bigcap_{n\ge1} I^n＝0$.

Is this true? Other proof are welcome.
Thank you for your help.

Answer 1

Your attempt seems correct but I am not certain if you obtained it without the result to be proven here.

I would just note that if $x\in \cap_j I^j$, then it maps $x$ to $(0,0,0,0,\ldots)$ via the canonical map into the product. Completeness would imply that the map is injective, so that $x=0$. One would then conclude the intersection is just $\{0\}$.