Mesh of iterated simplicial barycentric subdivision tends to zero

For points $x={\sum }_i{a}_i{v}_i$ and $y={\sum }_j{b}_j{v}_j$ in a simplex, $\mathrm{\parallel }x-y\mathrm{\parallel }\le {\sum }_{i,j}{a}_i{b}_j\mathrm{\parallel }{v}_i-v_j\mathrm{\parallel }\le {\max }_{i,j}\mathrm{\parallel }{v}_i-v_j\mathrm{\parallel }$, since all weights are nonnegative and sum to $1$. The maximum is attained by vertices, so it equals the diameter. A point has diameter zero; the definition of mesh assigns zero to a vertex-free complex without taking a diameter of the empty set.

For nonempty nested faces $F⊊G\subseteq \sigma$, $b_G=(\mathrm{\#}F\mathrm{/}\mathrm{\#}G)b_F+(1-\mathrm{\#}F\mathrm{/}\mathrm{\#}G)b_{G\setminus F}$. Thus $\mathrm{\parallel }{b}_G-b_F\mathrm{\parallel }\le (1-\mathrm{\#}F\mathrm{/}\mathrm{\#}G)\mathrm{diam}\sigma \le n\mathrm{/}(n+1)\mathrm{diam}\sigma$. Vertices of every subdivided simplex form such a chain, so the previous diameter calculation bounds its diameter by this factor. Iteration gives the asserted estimate. Since $0<n\mathrm{/}(n+1)<1$, its powers tend to zero. Zero-dimensional simplices remain points.

For any $x$ in the open or closed vertex star of $v$, some simplex contains both $x$ and $v$, so $\mathrm{\parallel }x-v\mathrm{\parallel }\le m(K)$. For two points $x,y$ in that star, the triangle inequality gives $\mathrm{\parallel }x-y\mathrm{\parallel }\le 2m(K)$, and taking the supremum proves the star bound. Finite weak realization topology agrees with the Euclidean topology, so these estimates use a compatible metric.