I need a little help regarding one of the basic properties of $\omega _1$.
Namely, I am supposed to show that for any $\alpha \in \omega_1$ there exists $\beta \in \omega_1\cap A$ such that $\alpha < \beta$.
Here $A$ stands for a set $A:=$ { $\alpha$ $ |$ $ \alpha \in Ord$ $\wedge$ $cof(\alpha)=\omega$ }.
What I am trying to show is that for any $\alpha \in \omega_1$ $\exists \beta \in \omega_1 \cap Lim$, where $Lim$ is the class of all limit ordinals such that $\alpha < \beta$.
This will be enough since every countable limit ordinal has cofinality $\omega$ which is not so hard to show.
My problem is, how to show that this defined $\beta$ indeed exists, i.e. $\beta$ being $Lim$.
Define $\beta = \sup \{ \alpha + n \mid n < \omega \}$. Then $\beta > \alpha$ and $\beta$ has cofinality $\omega$; moreover, it is less than $\omega_1$ since it is a countable union of countable sets, so is countable.