Finite simplicial approximation for maps of pairs

Use the barycentric homeomorphisms to view every subdivision as a triangulation of the same finite Euclidean polyhedron. The inverse images $f^{-1}({\mathrm{st}}_L(w))$ cover it, since the support of every image point is nonempty. They are open, and the source is compact metric. The Lebesgue-number lemma gives $\delta >0$ such that any nonempty set of diameter less than $\delta$ lies in one of these inverse images. For an empty source use the empty map for every $r$.

Choose $r_0$ with $2m({\mathrm{sd}}^{r}K)<\delta$ for every $r\ge r_0$, using the mesh estimate; in dimension zero $m=0$ already. Each vertex star in that triangulation is nonempty and has diameter at most $2m$. Hence for each of the finitely many vertices $v$ select $g(v)$ with $f(\mathrm{st}(v))\subseteq \mathrm{st}(g(v))$. This is only finite choice. The star criterion gives a simplicial map and a continuous common-carrier straight-line homotopy.

If $\sigma$ is a face of ${\mathrm{sd}}^{r}A$, its barycenter lies in $\mathrm{\mid }A\mathrm{\mid }$, so the support of its image under $f$ is a face of $B$. The star criterion puts all $g(v)$ for $v\in \sigma$ in this support; hence $g(\sigma )\in B$. At every point of $\mathrm{\mid }A\mathrm{\mid }$ the same carrier argument keeps the homotopy inside $\mathrm{\mid }B\mathrm{\mid }$. Pulling back to the abstract subdivided realization gives the stated homotopy of pairs to $f{b}_K^r$, for every $r\ge r_0$.