I've been reading up on the Gromov-Hausdorff metric, using the following books:
- A Course in Metric Geometry, by Dmitri Burago, Yuri Burago, and Sergei Ivanov
- Metric Structures for Riemannian and Non-Riemannian Spaces, by Misha Gromov
I'm particularly interested in learning the proof that the Gromov-Hausdorff space is complete. That is, in the space of (isometry classes of) compact metric spaces endowed with the GH metric, any Cauchy sequence is convergent. However, neither of these references seem to offer a clear proof of this fact, even though it is cited in other places (e.g. the wikipedia page or these papers).
As far as I can tell, this fact is not stated at all in the first source. In the second, Gromov says that it is "easy to see" this fact (p.78, remark $3.11 \tfrac{1}{2}_+$). Unfortunately, it is not easy to see for me 🙁
My thought is that it could be a consequence of the Gromov-Hausdorff precompactness theorem (theorem 7.4.15 in the first reference). However, the proof offered in "A course in Metric Geometry" extracts converging subsequences that don't seem like they should converge even if we're working with a Cauchy sequence.
Can anyone provide a proof or a reference to a proof?
EDIT: I just realized that using the Gromov-Hausdorff precompactness theorem should be enough: indeed, if a Cauchy sequence has a convergent subsequence, then it must be convergent. However, this means we need to prove that a Cauchy sequence is uniformly totally bounded, and it also seems like overkill. A direct proof is preferable for my purposes.
Best Answer
It's not just you: T. Tao left this as an exercise in one of his texts, and I did not find the construction all that easy. In any case, P. Petersen's Riemannian Geometry proves the completeness of Gromov-Hausdorff space as part of Proposition 11.1.8.