I am reading "Riemannian Geometry" by Petersen. In his proof for Gromov's precompactness theorem he looks at finite metric spaces with less than $N$ elements and distances between the elements of less than $D$ (let's call the collection of these finite metric spaces $C$). According to the proof these can be viewed as matrices $(d_{ij}) \in \mathbb{R}^{N \times N}$, where $d_{ij} = d(x_i, x_j)$, where $x_i, x_j$ are elements of the finite metric space. So these matrices are bounded by $D$, which gives a compact metric space, I guess. So we can find a finite covering for this compact matrix space, which gives a finite covering for the original finite metric spaces.
So this is only possible if the mapping from the collection of finite metric spaces $C$ to the space of compact matrices is continuous, right? In order to see if this mapping is continuous, we would have to assess the Gromov-Hausdorff distance between the finite metric spaces, right? Or is there any other way to see this analogy between the matrices and the finite metric spaces?