Isometric embeddings of finite metric spaces

It is proved that there exists a metric on the Cantor set such that any finite metric space with the diameter not exceeding 1 and the number of points not exceeding $n$ can be isometrically embedded into it. We also prove that for any $m,n \in \mathbb N$ there exists a Cantor set in $\mathbb R^m$ that isometrically contains all finite metric spaces embedded into $\mathbb R^m$, containing not more than $n$ points, and having the diameter not exceeding $1$. The latter result is proved for a wide class of metrics on $\mathbb R^m$ and in particular for the Euclidean metric.