The title is one of my homework assigments, (and the above greek letters are ordinal numbers) but this is the first one containing inequalities, and I dont really know how to do it. My guess is induction, and I think this might be a way to do it:
First we prove that $\alpha\leq \beta$ imply that $\alpha+\gamma \leq \beta +\gamma$:
Induction on $\gamma$. If $\gamma=0$ then the statement is trivial. Now assume that the statement holds for all $x<\gamma$. If $\gamma$ is a succesor, then there exists a $y$ such that $Sy=\gamma$, then:$\alpha+\gamma=S(\alpha+y)\leq S(\beta+y)=\beta+\gamma$If $\gamma$ is a limit ordinal, then:$\alpha+\gamma=\mbox{sup}(\alpha+x: x<\gamma)\leq \mbox{sup}(\beta+x: x<\gamma)=\beta+\gamma$and thus we have that $\alpha+\gamma\leq \beta+\gamma$. Then my idea is to prove that if $\gamma\leq \delta$, then we have that $\beta+\gamma\leq \beta+\delta$ and then the statement would follow.
My question: For this last part do we do induction with base case $\delta=\gamma$? and then do the same routine of considering cases when $\delta$ is a limit or a successor. Is there another way to tackle this problem?
Thanks!