I know that $(\omega +1 )^ \omega = \omega ^ \omega$ . I suppose because is
$(\omega +1 )^ \omega = \sup_{n<\omega} [ \omega ^n + \omega ^{n-1} ... +1 ] = \omega ^ \omega$. Is this correct?
Could someone explain to me simply why $\omega ^ \omega = \sup [ \omega ^n + \omega ^{n-1} + ... +1 ]$? Thanks a lot.
It's a kind of "lockstep" phenomenon. Here's a hint: note that for all $n$ we have $$\omega^n<\omega^n+\omega^{n-1}+...+1<\omega^{n+1}$$ and recall that $\omega^\omega=\sup\{\omega^n: n\in\omega\}$ by definition.