Proving Regarding Ordinal Numbers

Question

I am starting to learn order theory and the idea of ordinals. Here is the question that I have been struggling to answer for a long time: $\alpha$ is an ordinal number, and let $X$ be the set of positive integers $n$ for which $\alpha$ can be expressed as $\alpha = \beta + n$, where $\beta$ is an ordinal. How can I prove that $X$ has a maximal element if it is non-empty?

I would sincerely appreciate any help on this. Thanks!

Answer 1

HINT: Consider $\{\beta<\alpha:\alpha=\beta+n\text{ for some }n\in\Bbb Z^+\}$. I’ve expanded on that a little in the spoiler protected block below.

Specifically, consider the least element of that set and its associated $n$, assuming that the set is non-empty. (If the set is empty, what kind of ordinal is $\alpha$?)