- Define an equivalence relation on the set of infinite real sequences by: a~b iff a-b is bounded. For example, the equivalence class of 0 (the constant sequence 0,0,0,...) is the set of all bounded sequences; the class of n (i.e. 1,2,3,...) is the set of all sequences of the form "n + a bounded sequence" and so on. I wonder if there is a way to write a set of representatives to this relation (i.e. a set that consists of exactly one sequence from each class)? I know there are infinitely many classes (e.g. [n],[2n],[3n],... but also [n^2],[n^3],[ln(n)],...) and I know (by the axiom of choice) there exists a set of representatives, but I don't know if there is a way to build/describe such a set. I hope my question is clear. (I couldn't think of any way to build such a set and I'm not sure it's possible).
- Same question but for the relation "a-b is convergent".
- Same question but for the relation "a-b converges to 0". (I did notice that if I define this relation only on the set of convergent sequences then I have a way to describe a set of representatives- one for each limit).
I feel like the answer is negative for all of my 3 questions but I want to make sure and also hear your thoughts.
Thank you!