Let $k$ a field and $R= k[[X_1,X_2, ..., X_n]]$ the regular local ring of formal power series with maximal ideal $(X_1,X_2, ..., X_n)R$. How to prove that $R$ is a regular ring?
My attempts: $k$ is clearly regular, so firstly I tried to do it inductively, therefore to show following statement: If $A$ is local regular with max ideal $m_A$, then $A[[X]]$ is also local regular with obvious maximal ideal $(X)$.
But I don't know how to make here the induction step.