Consider the equivalence relation ~ on the set of all rational Cauchy sequence $\mathscr{C}$, and $\{x_n\}$~$\{y_n\}$ if $\{x_n-y_n\}$ tend to zero, $\{x_n\},\{y_n\}\in \mathscr{C}$. Now show that addition and multiplication are well defined on the quotient $\mathscr{C}/$~. I have no idea what it means to show these two operations are well defined, but after reading some online definitions, here are my attempts, Can anyone tell me whether or not I am correct? if not what I have to do to show it's well defined?
Suppose $x,y,x^\prime,y^\prime\in \mathscr{C}$ such that $x=x^\prime, y=y^\prime$, define $x+y=\{\{z_n\}\in \mathbb{Q}:\{z_n\}\text{~}\{x_n+y_n\}\},x_n\in x,y_n\in y$.
Then we have $x+y=\{\{z_n\}\in \mathbb{Q}:\{z_n\}\text{~}\{x_n+y_n\}\} $
$=\{\{z_n\}\in \mathbb{Q}:\{z_n\}\text{~}\{x^\prime_n+y^\prime_n\}\}$,by transitivity of the relation since $\{x_n+y_n\}$~$\{x^\prime_n+y^\prime_n\}$.
$=x^\prime+y^\prime$, hence addition is well defined.
For multiplication define $x\cdot y=\{x_ny_n\}$ and we have $x\cdot y=\{\{z_n\}\in \mathbb{Q}:\{z_n\}\text{~}\{x_ny_n\}\}=\{\{z_n\}\in \mathbb{Q}:\{z_n\}\text{~}\{x^\prime_ny^\prime_n\}\}=x^\prime\cdot y^\prime$,since $\{x_ny_n\}$~$\{x^\prime_ny^\prime_n\}$, hence multiplication is well defined.