I have two function $G$ and $F$ defined on ordinals and I know that
$$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I also know that $G(\omega)\subseteq F(\omega)$ and that $G$ and $F$ are increasing functions.
I have to prove that
If $\lambda <\omega$ then $G(\omega^\lambda)\subseteq F(\omega^{2(\lambda-1)}$).
If $\lambda=\omega^\alpha c$ where $\alpha\geq 1$, $1\leq c<\omega$, then $G(w^\lambda)\subseteq F(\omega^{\omega^\alpha(2c-1)})$
- If otherwise $c\geq \omega$, then $G(\omega^\lambda)\subseteq F(\omega^{\lambda 2})$
I've proved the first but the second and third elude my abilities.
About the second. I've tried by induction (setting $\alpha=1$ to try to see the general reasonment). But, assume the property true for $c$ and prove it for $c+1$: $$G(\omega^{\omega(c+1)})=G(\omega^{\omega c+\omega})=\cup G(\omega^{\omega c+n})$$
Now I thinked to do induction on $n$ but it seems to me that things go worse.
Have you some hints for me?