Reading through my textbook I feel like I understand the justification behind constructing the set of real numbers as a quotient set of Cauchy sequences in Q but I'm confused about how this would actually be done.
The claim in the book is this should be able to be done with some Cauchy sequence $(x_n\in \mathbb{R})_{n \in \mathbb{N}}$, with each $x_n$ having a representative $x_n = [(x_{n,m}\in \mathbb{Q})_{m\in\mathbb{N}}]$ such that $(x_n,n \in \mathbb{Q})_{n\in\mathbb{N}}$ is Cauchy in $\mathbb{Q}$ but $lim_{n\in\mathbb{N}}x_{n,n}\neq lim_n x_n$.
I just can't figure out any sequences that would satisfy these properties. Don't need a rigorous proof but are there any easy examples?
For a very simple example, let $x_n=0$ for all $n$, and $x_{n,m}=0$ if $m\neq n$ and $x_{n,n}=1$ for all $n$. Then $(x_n)$ converges to $0$ but $(x_{n,n})$ converges to $1$.