How to use induction to prove that if $α+β=β$, then for every $n<ω,α⋅n\le β$ and hence $\alpha\omega\le \beta$?

Question

Here $\alpha,\beta$ are ordinals.

I know that it can be done by induction, but searching in my textbook I cannot find how exactly can I deal with it. I am completely new to ordinal arithmetic, so may I please ask for an explicit proof by induction? Thanks a lot.

Answer 1

Hint: use the following properties, and if necessary (i.e. if you don't know them) prove them:

$+$ is associative; $+$ is commutative on finite ordinals; $\cdot$ is distributive over $+$ on the left, i.e. $\alpha\cdot(\beta+\gamma) = \alpha\cdot \beta + \alpha\cdot\gamma$ for all ordinals $\alpha, \beta, \gamma$; if $\alpha+\beta = \beta$, then $\alpha \leq \beta$