Limit ordinal properties

Question

Given $\alpha \in Ord$ and $\gamma \in Lim$

where $Ord$ is the class of all ordinal numbers and $Lim$ class of all limit ordinals

(i.e. those which are not succesors),

I was wondering how to prove that in that case $\alpha$ + $\gamma$ $\in Lim$ too.

Answer 1

Note that an ordinal, as a linear order, is a limit ordinal if and only if it does not have a last element.

Next, note that a linear order does not have a last element if and only if there is a tail segment without a last element.

Finally, the ordinal sum $\alpha+\gamma$ can be defined as an ordinal with an initial segment of order type $\alpha$ and a tail segment with order type $\gamma$.