Tail suprema land in the scale subspace

Each $h_{\xi }$ has exactly one scale index ${\alpha }_{\xi }$ with $h_{\xi }{=}^{\ast }{f}_{{\alpha }_{\xi }}$. Existence is F1. If two distinct indices worked, put the smaller first; scale strictness outside a finite set would give $h_{\xi }(n)<h_{\xi }(n)$ at all but finitely many coordinates. The infinite $B$ has a coordinate outside their finite union, a contradiction. For $\xi <\eta$, the supplied strict inequality on the cofinite tail excludes ${\alpha }_{\eta }\le {\alpha }_{\xi }$: equality would imply eventual equality of the two points, while a strict reverse index inequality would imply $h_{\eta }{<}^{\ast }{h}_{\xi }$. Either conflicts with the strict tail inequalities outside finitely many coordinates. Thus $({\alpha }_{\xi })$ is strictly increasing.

Set $T=\{n\in B:n>k\}$. For every $\xi <{\mathrm{\aleph }}_m$ there is a successor $\xi +1<{\mathrm{\aleph }}_m$, since the infinite cardinal ${\mathrm{\aleph }}_m$ is a limit ordinal (also its cofinality exceeds one by F3–F4). For $n\in T$, $h_{\xi }(n)<h_{\xi +1}(n)\le {\mathrm{\aleph }}_n$, so $h_{\xi }(n)<{\mathrm{\aleph }}_n$. Hence the restrictions $h_{\xi }\upharpoonright T$ are members of the actual strict product, not just the inclusive-top product. They are pointwise increasing by hypothesis and eventually equal to $f_{{\alpha }_{\xi }}\upharpoonright T$ by step 1.1. Thus all hypotheses of F2, including $1\le m\le k$, hold. F2 and A1 give $\delta ={\sup }_{\xi }{\alpha }_{\xi }<\lambda$, $\mathrm{cf}(\delta )={\mathrm{\aleph }}_m$, and $g\upharpoonright T{=}^{\ast }{f}_{\delta }\upharpoonright T$ with $g(n)<{\mathrm{\aleph }}_n$ and $\mathrm{cf}(g(n))={\mathrm{\aleph }}_m$ on $T$.

Define $h(n)={\mathrm{\aleph }}_n$ for $n\in B$ with $n\le k$, and $h(n)=g(n)$ for $n>k$. On the finite prefix, F3 gives uncountable cofinalities ${\mathrm{\aleph }}_n\le {\mathrm{\aleph }}_k$; on the tail, step 2.1 gives cofinality ${\mathrm{\aleph }}_m\le {\mathrm{\aleph }}_k$. Thus every coordinate cofinality lies strictly between $\omega$ and ${\mathrm{\aleph }}_{k+1}$, and $h\in X_R(B)$. Step 2.1 and the finite prefix imply $h{=}^{\ast }{f}_{\delta }$, so F1 gives $h\in X$. It agrees with $g$ on every required tail coordinate and differs, if at all, only on the finite prefix. QED.