Let $A$ be a commutative ring with unity, $M$ an $A$-module and $I$ an ideal of $A$.
Now it is fairly obvious that we invoke the usual definition of a Cauchy sequence of $M$ in the $I$-adic topology as follows : a sequence $(x_n)_{n \ge 1}$ of elements in $M$ is Cauchy $\iff$ for every positive integer $r$, there exists $n_0$ such that $x_{n+1}-x_n \in I^rM ,\forall n > n_0$. Similarly, completeness can then be expressed as saying that a Cauchy sequence has a unique limit.
But then Matsumura remarks in his 'Commutative Ring Theory' :
As one can easily check, to say that $M$ is complete for the $I$-adic topology, is equivalent to saying that for every sequence $(x_n)_{n \ge 1}$ of elements of M satisfying $x_i -x_{i+1} \in I^iM ,\forall i$, there exists a unique $x \in M$ such that $x - x_i \in I^iM ,\forall i$.
I really can't understand how he is proving this equivalence! Like, how is he regulating the same $i$ for the sequence elements $x_i$ as well as the open set from the filtration $I^iM$ . Is it a stronger version or is indeed equiavelent to the notions as discussed on top!
Would really appreciate clarifications and proofs. Thanks for help.
Reference: Bottom of Page 57, 'Commutative Ring Theory' - Matsumura
The completion of $M$ is the inverse limit of the system $M/IM\xleftarrow{f_1}M/I^2M\xleftarrow{f_2}\cdots$ or equivalently
$$\hat{M}^I=\{(x_n)_{n\geq 1}\in\prod_{n=1}^\infty M/I^nM\mid\forall n\geq 1,\ f_n(x_{n+1})=x_n\}.$$
Let $\pi_n:M\to M/I^nM$ be the canonical surjection. We say $M$ is complete if the canonical map $\varphi: M\to \hat{M}^I$ given by $\varphi(x)=(\pi_n(x))_{n\geq 1}$ is an isomorphism. If $(x_n)_{n\geq 1}$ is a Cauchy sequence in $M$, then clearly $(\pi_n(x_n))_{n\geq 1}\in \hat{M}^I$. The surjectivity of $\varphi$ implies this Cauchy sequence has a limit, and the injectivity implies this limit is unique. Conversely, any element $(x_n)_{n\geq 1}\in \hat{M}^I$ lifts to a Cauchy sequence $(\tilde{x}_n)_{n\geq 1}$ in $M$ where $\pi_n(\tilde{x}_n)=x_n$ for all $n$, so the existence of a limit for any Cauchy sequence implies $\varphi$ is surjective and the uniqueness of such limits implies $\varphi$ is injective.