How can we show that $\omega_1$ is a regular cardinal?

Question

A cardinal $\kappa$ is regular if and only if there is no $\lambda<\kappa$ for which there is a function $f:\lambda\rightarrow\kappa$ with range cofinal in $\kappa$. How can we see in ZFC that $\omega_1$ is a regular cardinal?

Answer 1

HINT: If $A$ is a countable set of countable ordinals, then $\sup A=\bigcup A$ is a countable ordinal.