(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 $a \in A, a\geq s$?
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There is a counterexample in the long line L. It is totally ordered and every sequence has a limit in L. see the following: 


Keyword: cofinality. 

