Question

Does it help in any way, to think of sequences as countable subsets of the Real numbers? Does this way of thinking take us anywhere important?

Answer 1

It ultimately leads to the conclusion that there are multiple sizes of infinity. Cantor used a similar line of thought to actually come to that and more conclusions about infinity. Well worth studying, it's not too hard but very deep and profound.

Answer 2

Does this thought lead to any theorem, or any understanding about the concept of limit?