Countable subset of an unbounded chain

Question

Let $C$ be a chain (i.e. a total order set) with no maximal element. Is always possible to find a countable subset of $C$ with no upper bound?

Answer 1

Let $C=\omega_1$, the first uncountable ordinal. Then all elements of $C$ are countable ordinals. Let $A\subseteq C$ be countable. Then $\bigcup A$ is also a countable ordinal, hence is in $C$ and is an upper bound of $A$.