How to prove $\gamma < \omega_1$?

Question

Let $\gamma=\sup\{\xi_n\}=\{\xi_n<\omega_1: n\in \omega\}$. How to prove $\gamma < \omega_1$?

Thanks for your help.

Answer 1

Since $\gamma=\bigcup_{n\in\omega}\xi_n$, it’s the union of countably many countable sets and is therefore countable; thus, it must be less than $\omega_1$.