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<br /> <br /> = \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}