For ordinals $\alpha$, $\beta$, $\gamma$, if $\gamma$ is a limit ordinal and $\beta = \sup\{\beta_{\delta}:\delta<\gamma\}$, why does below expression hold,
$$\alpha+\beta=\alpha + \sup\{\beta_{\delta}:\delta<\gamma\}=\sup\{\alpha+\beta_{\delta}:\delta<\gamma\}$$
Simply what I am asking is why,
$$\alpha+\beta=\sup\{\alpha+\beta_{\delta}:\delta<\gamma\}$$
holds? I tried to prove it by considering another $\sup$ like $z$ then trying to prove that the $\sup$ we've found is less that $z$. I don't know why and where in proof $\gamma$ is limit ordinal is required?!
Edit I simply want to prove continuity property of ordinal addition. There was a question asked about what continuity of ordinals is here.