Show that a particular set is not a limit ordinal (aim is to define ordinal subtraction)

Question

Let $\alpha$ and $\beta$ be two ordinals with $\beta \leq \alpha$. Define

$$ X:= \{\gamma \in \alpha^{+} : \beta + \gamma \leq \alpha\}.$$

I have shown this is an ordinal. Now I need to show it isn't a limit ordinal, but I'm stuck.

Many thanks for your help.

Answer 1

HINT: Assume to the contrary, and use the definition of $\beta+\delta$ when $\delta$ is a limit ordinal, to show that $\beta+\delta\leq\alpha$ as well to obtain a contradiction.