I'm trying to understand the general concept of completing arbitrary normed fields.It's a long question because I tried to include as much context as needed. What I learned so far from Katok's p-adic Anaysis :
We have $(F,||\cdot||)$. The first step is to introduce the set of all Cauchy Sequences $\{F\}$ on $F$. After that we introduce $\hat{F}=\{F\}/P$, that is modulo, P, where P is the set ( more precisely an ideal ) of all the null sequences on $F$.
That is a field. We supply the norm $$ ||A|| = \lim\limits_{n \rightarrow \infty} ||a_n||$$ where $A\in \hat{F}$ with $\{a_n\} \in A$.
It's complete, and the set $\hat{F}_0$ is dense in $\hat{F}$, where $F \rightarrow \hat{F}_0$ is an isometry. $\hat{F}_0$ is the set of all the equivalence classes that have a representative which is a constant cauchy sequence of elements in $F$.
That means since elements of $\hat{F}$ are equivalence classes of Cauchy Sequences, we can identify the elements of $F$ by looking at the equivalence class of the constant cauchy sequence $(\hat{a})$ for every $a\in F$.
So far so good, here comes the problem
The book states:
The operations on $\hat{F}$ are extended by continuity from $F$, i.e. if $A = \lim\limits_{n \rightarrow \infty} (\hat{a}_n)$ and $B = \lim\limits_{n \rightarrow \infty} (\hat{b}_n)$, then
$$ A+B = \lim\limits_{n \rightarrow \infty} (\hat{a}_n+\hat{b}_n)$$ $$A\cdot B = \lim\limits_{n \rightarrow \infty} (\hat{a}_n \cdot \hat{b}_n)$$
By $\hat{a}_n$ they mean the equivalence class of the constant sequence $a_n = a$.
But how is this consistent with the already well defined operations on $\hat{F}$?
Namely that for $A=(\{a_n\}),B \in \hat{F}$, we have
$$ A+B = (\{a_n+b_n\})$$
and the same for multiplication? Where do the limits suddenly come from? What even is the limit of an equivalence class of a constant cauchy sequence? Is it $\lim\limits_{n \rightarrow \infty} (\hat{b}_n) = (\hat{b_n})$?
The notation $A = \lim a_n$ is a bit misleading. What it is meant: if $A$ is represented by the Cauchy sequence $(a_n)$, and $B$ by the Cauchy sequence $b_n$, then $A\cdot B$ is represented by the Cauchy sequence $a_n b_n$. This could be all streamlined if you took the ring of Cauchy sequences from $F$, the ideal of null sequences, and the factor ring. Then it looks more or less like defining an operation on a ring of form $\mathbb{Z}/n$, by representatives.