Consider the set of upper bounds of the given family of ordinals. Since the class of ordinals is well-ordered, and since this set is by assumption non-empty, there exists a minimal element in this set, which would be the supremum.
Is this correct?
This seems like a trivial case, but the book (p. 84, in Russian) I'm using builds an, in my opinion, way more involved proof, hence my question
It does look that easy to me. In fact, if you are using von Neumann ordinals, you can just take their union (since the collection is bounded, the union makes sense), and that will be the supremum.