limits of sequences of topological rings Let $A$ be a ring and $I$ an ideal of $A$ such that $A$ is complete in the $I$-adic topology. Let $a \in I$. Then the sequence $y_n=1-a+a^2-a^3+\cdots+(-1)^n a^n$ converges in $A$. By definition of completeness, this means that there exists a unique element $x \in A$ such that 
$y_n - x \in I^n, \, \, \, \forall n$. 
Question: Why does that imply that we can view the infinite series $1-a+a^2-a^3+\cdots$ as an element of $A$? In particular i encountered the argument that because $(y_n)$ converges, then 
$1-a+a^2-a^3+\cdots$ is the inverse of $1+a$, hence $1+a$ is a unit. 
Reference: Matsumura's Commutative Ring Theory proof of theorem 8.2.

The series $1-a+a^2-a^3\dots$ is just suggestive shorthand for $x\in A$. Take a look at what it means in terms of limits (in the way we are accustomed to doing for sequences of real numbers.)
If you compute $y_n(1+a)=1+(-1)^{n+1}a^n$, then
$$
x(1+a)=\lim_{n\to \infty} y_n(1+a)=\lim_{n\to \infty} 1+(-1)^{n+1}a^n=1+\lim_{n\to \infty}(-1)^{n+1}a^n
$$
But since $a\in I$, the limit on the far RHS is $0$. (This is because $\pm a^n\in I^n$ for each $n$, and $\cap I^n=\{0\}$.)
