Regarding the above question, the solutions require creating classes of sequences with representative sequences.
How are those sequences constructed?
How is it possible to generate a "list" of the classes and representatives if the sequences are made of real numbers which implies the set of sequences is uncountable?
Thanks in advance for any answers.
It is adequate for the question to define sequences as functions with domain $\mathbb{N}$. So a sequence of reals is just a function $\mathbb{N}\rightarrow\mathbb{R}$.
You are correct that there are uncountably many of them, but if you accept constructing $\mathbb{R}$ then why should the collection $\mathbb{R}^{\mathbb{N}} = \{f \ : \ f:\mathbb{N}\rightarrow\mathbb{R}\}$ be a problem?
They then go on to define an equivalence relation on $\mathbb{R}^{\mathbb{N}}$ by $f \sim g$ iff $\exists n \forall m >n f(m)=g(m)$ (this is called tail equivalence). They then pick a representative from each class, and this is where choice is used, it wasn't used before.