I find myself unable to solve question 24.1 of T. Jech's *Set Theory*:



> 
> If $\beta<\omega\_1$ and if
>  $2^{\aleph\_{\alpha}}\leq\aleph\_{\alpha+\beta}$
>  for a stationary set of $\alpha$'s,
>  then
>  $2^{\aleph\_{\omega\_1}}\leq\aleph\_{\omega\_1+\beta}$.
> 
> 
> [By induction on $\beta$: If
>  $\varphi(\alpha)\leq\beta$ on a
>  stationary set, then
>  $||\varphi||\leq\beta$.]
> 
> 
> 


I am unable to prove the hint (in brackets). Any hints for the hint?



Set Theory exercise.
Recall the definitions (24.1) and the consequence of the proof of Lemma 24.2, which states that $\|\varphi\|=\sup\lbrace\|\psi\|+1\mid\psi<\varphi\rbrace$.


Then we have that $\|\varphi\|=0$ if and only if $\varphi(\alpha)=0$ on a stationary set. Proceed by induction; suppose this claim is true for all $\delta<\beta$ and let $\varphi$ be such that $\varphi(\alpha)\leq\beta$. Clearly if $\varphi(\alpha)\leq\delta<\beta$ then we are done, so suppose that $\varphi(\alpha)=\beta$ for a stationary set.


So if $\psi<\varphi$ we have that on a club set, $\psi(\alpha)<\beta$, and therefore has rank of at most $\delta<\beta$. So the rank of $\varphi$ is at most $\beta$.


