If $\kappa$ is a singular cardinal, a *scale for $\kappa$* consists of an increasing sequence $\langle \kappa_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\langle f_\alpha : \alpha < \kappa^+ \rangle$ that is linearly ordered and dominating in the partial order of $\prod_{i<\mathrm{cf}(\kappa)} \kappa_i$ by $f \leq g$ iff $|\{ i : f(i) >g(i) \}| < \mathrm{cf}(\kappa)$. Shelah proved that scales exist for every singular cardinal.

A point $\alpha < \kappa^+$ is called *good* for a scale on $\kappa$ if there is an unbounded $A \subseteq \alpha$ and an $i < \mathrm{cf}(\kappa)$ such that for all $j > i$, $\langle f_\beta(j) : \beta \in A \rangle$ is a strictly increasing sequence of ordinals.

The following seems to be folklore:

**Theorem (ZFC):** For every scale $\vec F$ for $\aleph_\omega$, there is a club $C \subseteq \aleph_{\omega+1}$ such that every point in $C$ of cofinality at least $\aleph_4$ is good.

**Question 1:** Is there a proof of this in the literature? Or can one neatly construct a proof from some lemmas stated in the literature?

**Question 2:** How does this generalize to larger singular cardinals?