Having a quick problem with a line in Grillet's Abstract Algebra, Prop. VI.9.2 on p. 267-8 on the topic of Filtrations and Completions. He says that the ideal
$$\widehat{\mathfrak{a}}_j = \{ (x_1 + \mathfrak{a}_1 , \dots , x_i + \mathfrak{a}_i , \dots ) \in \widehat{R}_{\mathcal{A}} \ | \ x_j \in \mathfrak{a}_j \}$$
is the kernel of the morphism $\widehat{R}_{\mathcal{A}} \rightarrow R/ \mathfrak{a}_j$ defined by $(x_1 + \mathfrak{a}_1 , \dots , x_i + \mathfrak{a}_i , \dots ) \mapsto x_j + \mathfrak{a}_j$.
Calling that morphism $\theta$, I have of course no problem seeing that $\widehat{\mathfrak{a}}_j \subseteq \ker \theta$, but I cannot quite see how $\ker \theta \subseteq \widehat{\mathfrak{a}}_j$. It is of course clear that $x_j - x_{j-1} \in \mathfrak{a}_{j-1}$, but I don't see how you go from knowing that $x_j \in \mathfrak{a}_j$ into concluding that $x_{j-1} \in \mathfrak{a}_j$ with that piece of information.
Anyone care to help me?