Does every totally ordered set contain an unbounded countable subset. In other words: If S is a totally ordered set, can we find a countable subset A, such that for every $s \in S$, there is a $a \in A, a\geq s$?

(Edit: The first formulation is wrong. See the second answer) Does every totally ordered set contain an unbounded countable subset. In other words: If S is a totally ordered set, can we find a (edit: at most) countable subset A, such that for every $s \in S$, there is a $a \in A, a\geq s$?