Cantor normal form multiplication

[daniel_i_l]

Let's say that a is an ordinal and it's cantor normal form is:
$$a = {\omega^{\beta_1}}c_1 + {\omega^{\beta_2}}c_2 + ...$$
I read that
$$a \omega = {\omega^{\beta_1+1}}$$
But I couldn't find a proof anywhere.
Can someone give me a source or point me in the right direction so that I can prove it myself?
Thanks.

 

[JSuarez]

The $$c_{i}'s$$ are positive integers, so $$c_{i}\omega=\omega$$ and:

$$\left(\omega^{\beta_{i}}c_{i}\right)\omega=\omega^{\beta_{i}}\omega = \omega^{\beta_{i+1}}$$

But the $$\beta_{i}$$ are in descending order, so their sum is equal to the largest element, which is $$\omega^{\beta_{1}+1}$$