This definition of ordinal adddition is taken from Kenneth Kunens "Set Theory: An Introduction to Independence Proofs":
$\alpha + \beta = \text{type}(\alpha \times \{0\} \cup \beta \times \{1\},R)$ where
$$ R = \{ \langle \langle \xi,0 \rangle,\langle \eta,0 \rangle \rangle : \xi < \eta < \alpha\} \cup \{ \langle \langle \xi,1 \rangle, \langle \eta,1 \rangle \rangle : \xi < \eta < \beta \} \cup \left[(\alpha \times \{0\}) \times (\beta \times \{1\})\right]. $$
Now, lemma 7.18.(5) gives us that if $\beta$ is a limit ordinal, then $\alpha + \beta = \sup\{\alpha + \xi : \xi < \beta\}$. I am a bit confused as to how to prove this using the definition given above. I am not quite comfortable with ordinals yet, and would be happy for hints/suggestions/full solutions.
Remark: Note that I am interested in proving this from Kunens definition; so unless I am mistaken, I don't want a proof in terms of induction, of which I have already seen (unless there is some way to apply induction in the context of this definition of ordinal addition).
Remark 2: I am quite comfortable with the definition itself, but I am probably missing some piece for how to apply it here. For example, do I have to consider the cases when $\alpha$ is a limit ordinal, and when $\alpha$ is a successor ordinal, separately?
