I'm attempting to learn about the construction of real numbers $\mathbb{R}$ by completion of the rational numbers $\mathbb{Q}$. The notes I am following do this by defining the set of real numbers as cosets of Cauchy sequences. However, when proving this set to be complete I ran into some notation that I had not seen before. Here is some introduction to where I am:
Let $Y$ be the cosets, as mentioned, and let $d$ be a metric hereon. I wish to prove that the metric space $(Y,d)$ is complete. To this end, my notes define a family of Cauchy sequences in $\mathbb{Q}$ as:
$(x_n^{(k)})$, k=1,2,3,...
And a Cauchy sequence in Y:
$\lbrace [(x_n^{(k)})]\rbrace_{k=0}^{\infty}$
I have trouble decoding what is meant by these two notations; that is, I am not sure how to visualize (or even understand) what is meant by the definitions. As a bonus I would like to ask what a 'family' is in this context. Thank you very much,
kasp9201