# Silver's Theorem and $\lambda^{\aleph_0} = \lambda$ for regular $\lambda \geq \kappa$

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.

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, $$\kappa^{\aleph_0}\leq(\kappa^+)^{\aleph_0}=\kappa^+$$.