Prove If $\alpha$ is a non zero limit ordinal then ${\omega}^{\alpha}$ is a limit ordinal

Question

My attempt so far is that if ${\omega}^{\alpha}$ = sup(${{\omega}^{\alpha}}$)

then ${\omega}^{\alpha}$ is a limit ordinal.

For ${\omega}^2$ case: ${\omega}^2$= $\omega$ $\cdot$ $\omega$ = sup{${\omega}{\cdot}n {\mid}n<{\omega}$}

And then, for ${\beta} + 1 $ $\leq$ ${\omega}^2$

which shows that the above is a limit ordinal.

But I am having trouble for showing ${\omega}^{\alpha}$ case.

How should I proceed from here?

I am very new to this set theory area, and stuck on this question for hours, thanks for any help or guidance for this problem win advance.