An upper bound of a countable set of countable ordinals

Question

Let $X$ be a countable set of countable ordinals. Is there any upper bound of this set under $\omega_1$? In other words, can we find an ordinal $\alpha<\omega_1$ which is larger than any elements of $X$?

Answer 1

HINT: If $X$ is a set of ordinals, then $\bigcup X=\sup X$. If $X$ is a countable set of countable ordinals, then what is the cardinality of $\bigcup X$?

(You have to use the axiom of choice for this, though!)