Applying the Galvin-Hajnal Theorem?

Question

Working my way through chapter 24 of Jech's Set Theory, but progress is painfully slow. I'm now stuck on question 24.3:

If $2^{\aleph_{\alpha}}\leq\aleph_{\alpha+2}$ holds for all cardinals of cofinality $\omega$, then the same holds for all singular cardinals.

It seems like the proof should be similar to that of Theorems 8.12 and/or 8.13, but quite frankly I'm utterly lost. Any hints?

Answer 1

Converting my comment to an answer:

Recall the proof of Silver's theorem, the behavior at singular cardinals with uncountable cofinality is determined on a stationary subset, and the cardinals with countable cofinality make such stationary subset.