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The well-known Silver's theorem asserts the relationship between Singular Cardinal Hypothesis $\mathsf{SCH}$ and cofinality:

Let $\kappa$ be a singular cardinal such that $\operatorname{cf}(\kappa) > \omega$. If $2^\alpha = \alpha^+$ for a stationary subset of cardinals $\alpha < \kappa$, then $2^\kappa = \kappa^+$.

In page 5 Assaf Rinot's notes titled "Surprisingly Short", Rinot wrote:

By a celebrated result of Silver from [7], to show that $\mathsf{SCH}$ holds above a cardinal $\kappa$, it suffices to prove that $\lambda^{\aleph_0} = \lambda$ for all regular $\lambda \geq \kappa$.

[7] is Silver's original paper "On the singular cardinal problem", which states exactly what I wrote above on Silver's theorem.

I'm not sure how Rinot's claim implies Silver's theorem, and I would like some help guiding me so.

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Recall Hausdorff's formula, $(\kappa^+)^{\aleph_0}=\kappa^+\cdot\kappa^{\aleph_0}$.

So if it is true that for every regular cardinal, $\kappa^{\aleph_0}=\kappa$, then for every singular cardinal of countable cofinality, $\kappa^{\aleph_0}\leq(\kappa^+)^{\aleph_0}=\kappa^+$.

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