As you say, the $\leq$ inequality follows from the fact that $\forall \beta \in A, \alpha + \beta \leq \alpha + \sigma$, where $\sigma := \sup A$.
For the other inequality, you can argue as follows: if $\sigma \in A$, then the claim is trivial to prove, so assume that $\sigma \not\in A$. Then ne...
Thank you so much for your answer! It seems to me that you meant $\alpha + \sigma = \alpha + \underset{\eta < \sigma}{\lim}\eta$ rather than $\underset{\eta < \sigma}{\lim}\alpha+\eta$, and $ \alpha + \underset{\eta \in A}{\lim}\eta$ rather than $\underset{\eta \in A}{\lim} \alpha + \eta$. On the basis of your answer, I have presented a detailed proof and posted it as an answer below. If you don't mind, please have a look at it. Thank you so much!
Thank to your elaboration, I got what that sentence's meaning. But I think the author's proof is not sufficient and not complete. Although he has proved that all ordinals less than $\sup\limits_{β∈A}(α+β)$ is also less than $α+\sup\limits_{β∈A}(β)$, he has done NOTHING in proving the converse direction that all ordinals less than $α+\sup\limits_{β∈A}(β)$ is also less than $\sup\limits_{β∈A}(α+β)$. As a result, he can not claim that Therefore the ordinals less than $\sup\limits_{β∈A}(α+β)$ and the ordinals less than $α+\sup\limits_{β∈A}(β)$ are the same. Thus he has not completed the proof.
@LeAnhDung, about the second comment, you are right! But the converse direction is easier, as you yourself noted. Maybe the author should have explicitely said that, anyway.
About the first comment, $\alpha + \sigma =\underset{\eta < \sigma}{\lim} \alpha + \eta$ by definition of ordinal addition for limit ordinals. (Of course, it is also equal to $\alpha + \underset{\eta < \sigma}{\lim} \eta$)
Honestly, I have not been able to prove the converse direction. I quite don't understand what you meant by But the converse direction is easier, as you yourself noted. Could you explain how to prove that all ordinals less than $α+\sup\limits_{β∈A}(β)$ is also less than $\sup\limits_{β∈A}(α+β)$?
Discussion between Guillermo Mosse an…
Discussion between Guillermo Mosse an…