User:Omlett71/Sandbox/Equivalence Relation on Rational Cauchy Sequences

Necessary Lemmas
Lemma 1

Any Cauchy sequence $x$ has an upper bound $M^+$ and a lower bound $M^-$ such that $M^- \le x_i \le M^+$.

Let $N \in \N$ satisfy for Cauchy sequence $x$ $n,m>N \implies |x_n-x_m| < 1$. Then in particular, $n>N \implies |x_n-x_N|<1$ and so if we define $M^{-*}:=x_N-1$ and $M^{+*}:=x_N+1$, these are lower and upper bounds respectively on all but a finite amount of entries. Setting $M^-:=\min\{M^{+*},\min\{x_i|i<N\}\}$ we have lower and upper bounds for the whole sequence, and as a direct consequence

Ring structure of Rational Cauchy Sequences
We can define a ring on the space of Rational Cauchy sequences by the operations of pointwise addition and multiplication, denoted by $\oplus$ and $\otimes$ respectively.

These operations are associative and commutative as a direct consequence of the same properties for addition and multiplication over $\Q$, and multiplication distributes over addition similarly.

However, it is necessary to show that these operations are algebraically closed, as is the operation of taking an additive inverse.

If we have Cauchy sequences $x$, $y$, then $z=x\oplus y$ has $z_i = x_i+y_i$. Because $x$,$y$ are Cauchy, for any $\epsilon \in \Q_{>0}$ there exist $N_x, N_y$ satsifying $n,m>N_x \implies |x_n -x_m| < \epsilon/2$ and $n,m> N_y \implies |y_n-y_m| < \epsilon/2$. Letting $N =\max\{N_y,N_x\}$, we then have that $n,m>N \implies |z_n - z_m| = |(x_n - x_m) + (y_n - y_m)| \le |x_n-x_m| + |y_n - y_m| < \epsilon/2 + \epsilon/2 = \epsilon$ by the triangle inequality, and thus $z$ is Cauchy.

Similarly, if we have Cauchy sequences $x$, $y$, then $z=x\oplus y$ has $z_i = x_i y_i$. Because they are Cauchy, we can define for any positive rational $\epsilon$ $N_x, N_y$ such that $n>N_x \implies |x_n-x_m|<1$ and $n>N_y \implies |y_n - y_m| < \epsilon$. Then if $N= \max\{N_x,N_y\}$, we have $n,m>N \implies |z_n-z_m| = |x_ny_n - x_my_m| = |x_ny_n - x_ny_m + x_ny_m - x_my_m| = |x_n(y_n - y_m) +y_m(x_n-x_m)|.

Lemma
We define the relation $\sim$ by $\langle x_n \rangle \sim \langle y_n \rangle \iff \forany \epsilon \in \Q_{>0}:\exists