TITLE: How to identify limit ordinals?
QUESTION [12 upvotes]: Definition: An ordinal number $\alpha$ is called a limit ordinal number if there is no ordinal number immediately preceding $\alpha$. 
Now my lecture notes say that $\omega, 2\omega, \omega^2, \omega^\omega$ are limit ordinal numbers whereas $\omega+3,2^\omega+5$ are not which is intuitively clear. But is there a characterization of limit ordinals that may work when proving the case with some rigor? Or is it "just look and identify" sort of a thing using the ordering of the family of ordinal numbers?

REPLY [7 votes]: You already have a characterization of limit ordinals. In any specific case you just have to verify that this characterization is fulfilled. Consider for example $\omega^2$:
$$
\omega^2 = \omega \cdot \omega = \sup \{ \omega \cdot n \mid n < \omega \}.
$$
Hence, if $\alpha < \omega^2$, there is some $n < \omega$ such that $\alpha < \omega \cdot n$. But then
$$
\alpha + 1 < \omega \cdot n + 1 < \omega \cdot (n+1) \le \omega^2.
$$
Thus $\omega^2$ is a limit ordinal.