I am reading Quasicoformally Homogeneous Domains by F. W. Gehring t And B. P. Palks
Let $D$ be a proper subdomain of $\mathbb{R}^n.$ Define the function $\rho.(x) = \frac{1}{dist(x, \partial D)}.$ For $x,y \in D$, define $$k_D(x,y)= \inf_{\gamma} \int_{\gamma} \rho ds.$$
I know that $$\lim_{y \to \partial D}k_D(x,y) = \infty \quad \text{ Eq : 2.3}$$.
Suppose $\{x_j\}$ is cauchy in metric $k_D$, I want to show that $\{x_j\}$ is Cauchy in Euclidean metric. Here is the screen shot : Here
I do not understand
If $\{x_j\}$ is Cauchy in the metric $k_D$, then $\{x_j\}$ is bounded in that metric and, consequently, lies in a compact subset of D in view of (2.3). This implies that $\{x_j\}$ is Cauchy in the euclidean metric.
I do not understand compactness mentioned withrespect to which metric. Can someone give more details.
The same question is posted here : https://mathoverflow.net/questions/432351/a-question-about-quasi-hyperbolic-m