How to prove the inverse limit $\lim_{n>0} R/\mathfrak{a}^{n}$ of a quotient of an adic ring is Hausdorff?

Question

I'm reading Bosch's Lectures on Formal and Rigid Geometry. And I'm have some problems. Let $R$ be an adic ring. How to prove that the completion of $R$ constructed by dividing the ring of all Cauchy sequences in $R$ by the ideal of all zero sequences is Hausdorff? And consider the inverse limit $\widehat{R}=\lim_{n>0} R/\mathfrak{a}^{n}$, where $\mathfrak{a}$ is the ideal of definition of $R$, how to prove that it is Hausdorff?

Answer 1

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Two different elements $x=(x_0,x_1,\ldots), y=(y_0,y_1,\ldots)$ will have some first component $n$ where $x_n\neq y_n$ in $R/\mathfrak{a}^n$. So, you can take $x+\mathfrak{a}^{n+1}$ and $y+\mathfrak{a}^{n+1}$ as neighborhoods of $x$ and $y$. They can't intersect since otherwise we would have $x_n-y_n\in \mathfrak{a}^n$. - plop