  It is shown that for any two compact metric spaces there exists an "optimal"
correspondence which the Gromov-Hausdorff distance is attained at. Each such
correspondence generates isometric embeddings of these spaces into a compact
metric space such that the Gromov-Hausdorff distance between the initial spaces
is equal to the Hausdorff distance between their images. Also, the optimal
correspondences could be used for constructing the shortest curves in the
Gromov-Hausdorff space in exactly the same way as it was done by Alexander
Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.03830, where it
is proved that the Gromov-Hausdorff space is geodesic. Notice that all proofs
in the present paper are elementary and use no more than the idea of
compactness.
