Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$.  Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with
$$
\left\{B\left(x_k,\frac1{n}\right)\right\}_{k=0}^{\mathbb{X}_n} \mbox{covers } X
\mbox{ and }
\#\mathbb{X}_n \mbox{is the $\frac1{k}$-covering number of $X$}.
$$
Fix some $x^{\star}\in X$ and consider the associated sequences of $1$-Lipschitz maps
$$
K_n:\,x\mapsto \left(d(x,x_n)-d(x_n,x^{\star})\right)_{x_n\in \mathbb{X}_n}.
$$

In the case where $(X,d)$ is a compact Riemannian manifold, then [this paper of Katz and Katz][1] (with un unpublished quantitative version [found here][2]) shows that $K_n$ approximate the [Kuratowksi embedding][3]
$$
K_{\infty}:x\mapsto \left(d(x,\cdot)-d(\cdot,x^{\star})\right) \in C_b(X,d).
$$

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What I mean is, if $(X,d)$ is compact (and possibly doubling as above) then can we always find sets $\{\mathbb{X}_n\}_{n=0}^{\infty}$ such that
$$
\lim\limits_{n \uparrow \infty}\,\max_{x\in X}\,|K_n(x)-K_{\infty}(x)|=0?
$$

*I'm assuming so, but I really can't see what the obstruction would be/why I can't find this result in the literature...*

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For instance, if $\emptyset\neq X'\subsetneq X$ and $X$ is a compact Riemannian manifold without boundary, then the construction should work by only considering points in $X'$.  Unless I'm missing something, this should work and I guess it should work for any doubling space by applying Assouad and ``replacing'' Lipschitz with Hölder?

  [1]: https://link.springer.com/article/10.1007/s10711-010-9497-4
  [2]: https://arxiv.org/pdf/1305.1529.pdf
  [3]: https://en.wikipedia.org/wiki/Kuratowski_embedding