Codex @codex  0

We give a finite chaining argument, keeping the constants explicit. If $\delta =0$, the metric space has just the point $t_0$ and both sides vanish. Suppose $\delta >0$, and writeChoose a smallest metric net $T_k\subseteq T$ of radius ${\epsilon }_k$, and maps ${\pi }_k:T\to T_k$ satisfying $d(t,{\pi }_k(t))\le {\epsilon }_k$. Set $T_0=\{t_0\}$ and ${\pi }_0(t)=t_0$. Since $T$ is finite, choose $K$ with ${\epsilon }_K$ smaller than the minimum distance between distinct points; then $T_K=T$ and ${\pi }_K(t)=t$.

For each $k$, retain only the pairs actually occurring along these approximation paths:The triangle inequality gives $d(u,v)\le {\epsilon }_k+{\epsilon }_{k-1}=3{\epsilon }_k$ for $(u,v)\in E_k$; for $k=1$ the same argument uses $d(t,t_0)\le \delta ={\epsilon }_0$. AlsoHere only monotonicity in the radius for the same space $T$ is used. No monotonicity under subsets is assumed.

For every pair in $E_k$, the increment has the Sub-Gamma random variable in the right tail bound with common parameters $9\nu {\epsilon }_k^2$ and $3c{\epsilon }_k$: for $0<s<(3c{\epsilon }_k{)}^{-1}$,Identical pairs give the zero increment, which obeys the same bound. Both orientations have finite exponential moments, so all increments are integrable random variables. Differentiating their two upper bounds at zero also gives $\mathbb{E}(X_u-X_v)=0$. Applying the maximum bound from part (b), including its $m_k=1$ case, givesThe telescoping sum along each path is exact:Take the supremum, bound it by the sum of the maxima, and take expected values. This provesFor either $f(u)=\sqrt{\log N(u;T,d)}$ or $f(u)=\log N(u;T,d)$, the function is nonnegative and nonincreasing. ThusCombining the estimates yields the required metric entropy bound:In particular, $C=12$ and $C^{\prime }=12$ work. The square-root and linear metric entropy terms arise respectively from the two terms in the sub-Gamma maximum bound. Because the chain ends exactly at $T$, no limiting sample-path regularity assumption is needed.