Let $C$ be the space of all Cauchy sequences of rational numbers. Let $X$ be the space whose elements are equivalence classes of elements in $C$. Define an equivalence relation on $C$ by
$$x\sim y \Longleftrightarrow \lim_{n\to\infty}|x_n-y_n|=0$$
Define, for elements $[x]\,,\, [y] \in X$,
$$[x]*[y]=[(x_n*y_n)]\,\,,$$ $$[x]+[y]=[(x_n+y_n)]\,\,,\, n \in\Bbb N$$
I proved the above operations are well-defined, and this showed commutativity and associativity. I need to show that distributivity of "*" over "+" holds.
Let $x, y, x', y'\in C$ , such that $x\sim x'\,\,,\, y\sim y'$.
In order to show distributivity, would I need to verify
$$\lim_{n\to\infty}|(x_n*y_n)(x_n'+y_n')-(x_n'*y_n')(x_n+y_n)|=0\,\,?$$ Because this is what I did, but I want to make sure it's the correct method.
Thanks.