What is a well-ordered set? Show that given any two well-ordered sets there is a unique order isomorphism between one and an initial segment of the other.

Show that for any ordinal $\alpha$ and for any non-zero ordinal $\beta$ there are unique ordinals $\gamma$ and $\delta$ with $\alpha=\beta \cdot \gamma+\delta$ and $\delta<\beta$.

Show that a non-zero ordinal $\lambda$ is a limit ordinal if and only if $\lambda=\omega \cdot \gamma$ for some non-zero ordinal $\gamma$.

[You may assume standard properties of ordinal addition, multiplication and subtraction.]