If $\alpha<\beta$, there exists a ordinal $\gamma$ such that $\alpha + \gamma = \beta$. Why? I want to prove the following proposition.
proposition: Let $\alpha,\beta$ be ordinals. If $\alpha<\beta$, there exists a ordinal $\gamma$ such that $\alpha + \gamma = \beta$.
I am trying to prove this by  transfinite induction. However, I have some trouble with the case $\beta$ is a limit ordinal.(I know the definition of the sum of a ordinal and a limit ordinal)
please give me some ideas.

As sets, $\alpha$ is a subset of $\beta$. Consider $C=\beta\setminus\alpha$, the set-theoretic difference. Then $C$ is a well-ordered set, having the same order type as some ordinal $\gamma$. Then $\beta=\alpha+\gamma$.
