I found the following proof that ordinal addition from the left preserves strict inequalities (see b)).
I am having trouble understanding the limit stage. The proof uses the inequality
$\gamma + \alpha < S(\gamma + \alpha) = \gamma + S(\alpha) \leq \sup(\gamma + \theta: \theta < \beta)$.
My thoughts are as follows: The first inequality simply follows from the definition of the successor of a set and the equality after that follows from the definition of addition of ordinals. The last inequality follows from the fact that $S(\alpha) < \beta$ such that $\gamma + S(\alpha)$ is an element of the set {$\gamma + \theta: \theta < \beta$}.
Is there any error in my line of argument? The fact that I don't need the induction hypothesis for the limit stage is generally a sign that something is wrong.
Thank you very much!
If $\beta$ is a limit, then pretty much by definition of a limit, ordinal addition with a limit, for all $\gamma$ we have that: $\alpha<\beta\to \gamma +\alpha<\gamma +\beta$. So you do not need the induction hypothesis. I will add that not using the induction hypothesis is not exactly uncommon.