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?