I was reading this paper http://www.math.umn.edu/~garrett/m/algebra/notes/14.pdf (p. 206, or p. 8 on a pdf) on Naive Set Theory on the proof that every ordinal is either the initial ordinal (empty set), a successor ordinal or a limit ordinal.
The proof in the test made sense except for one line that I did not understand the reasoning behind:
"The assumption $\beta < \alpha$ gives $\beta \cup \{\beta\} \le \alpha$."
An explanation would be greatly appreciated. Thanks.
Recall that $\beta+1=\beta\cup\{\beta\}$. Therefore $\beta\cup\{\beta\}$ is the successor ordinal of $\beta$.
Being a successor means that whenever $\beta<\alpha$ we have that $\beta+1\leq\alpha$. And so this holds.