I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of theorem 2.

The author defines the class $\mathcal{M}_n=\{$ $M $ compact riemannian manifold such that $dim(M)=n$ and $K\geq$ $5d \over 6$$\}$ where $K$ is the sectional curvature of $M$ and $d$ is a positive constant, the diameter of the Grassmannian $G(2,n)$, and a sequence $k_n=\sup_{M_n\in \mathcal{M}_n} (\sup _{m \in M_n} ||K_m||_{lip})$ where $||K_m||_{lip}$ is the Lipschitz constant of the function $K_m: G(2,n)\rightarrow \mathbb{R}$, the sectional curvature at point $m\in M$.

The author shows that $k_n$ is finite for every $n$ and then it seems that he deduces directly that $k_n \rightarrow 0$ thanks to a concentration theorem on Grassmanian (theorem 9).
I don't understand how we can apply this theorem to deduce the limit of the sequence.
Do you have any ideas?