(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 $a \in A, a\geq s$?



Unbounded countable subset
There is a counterexample in the long line L. It is totally ordered and every sequence has a limit in L. see the following:


[http://en.wikipedia.org/wiki/Long\_line\_(topology)](http://en.wikipedia.org/wiki/Long_line_%28topology%29)


