Ordinal addition being closed in infinite initial ordinals

Question

I am having real trouble proving that $\alpha + \beta < \omega_\gamma$ for any infinite initial ordinal $\omega_\gamma$ and any ordinals $\alpha < \omega_\gamma$ and $\beta < \omega_\gamma$. That is, that ordinal addtional is closed in any infinite initial ordinal. This is pretty easy to show for $\omega_0 = \omega$ since the addition of any two natural numbers is clearly still a natural number. However, I am not sure how to approach this for larger initial ordinals. I feel like cardinality has to play a role in this since pretty much all I know about infinite initial ordinals is in terms of cardinality. But there is no clear link between ordinal addition and cardinality. Any hints would be very helpful here!