Convergence of Subsequences in Gromov's Theorem A key theorem in metric geometry is Gromov's Compactness Theorem, which describes when a sequence of metric spaces has a Gromov-Hausdorff converging subsequence.
I'm not sure I follow how this works: does this mean that the sequence of metric spaces itself has to have a Gromov-Hausdorff limit in order for it to contain a Gromov-Hausdorff converging subsequence?  Also, what kinds of examples are there of converging subsequences in sequences of metric spaces apart from repeatedly rescaling a metric space?
 A: First of all, I highly recommend
Burago, D.; Burago, Yu.; Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society (AMS). xiv, 415 p. (2001). ZBL0981.51016.
specifically, Chapter 7, for a treatment of the GH-distance and convergence. If you want an example of a sequence of metric spaces which does not GH-converge but contains a convergent subsequence, the simplest example I know is:

*

*For $n$ odd, take the metric space $X_n$ consisting of a single point.


*For $n$ even  take the metric space $X_n$ consisting of two points $\{x, y\}$ such that $d(x,y)=1$.
Then the resulting sequence of metric spaces $(X_n)$ will not GH-converge, but will contains two convergent subsequences (one is $(X_{2k-1})$ and the other is $(X_{2k})$).
If you want more interesting examples (not obtained by a rescaling):
a. Take the sequence of metric spaces $X_n$ which are ellipses in the plane
$$
\{(x,y): a_n x^2+ y^2=1\}, a_n= 1+\frac{1}{n}. 
$$
(Take the restriction of the planar distance function to define the metric on these ellipses.) The limit of this sequence will be the unit circle.
b. Take the sequence of Euclidean rectangles of the shape $a_n\times b_n$, where $a_n\to 1, b_n\to 0$. This sequence will converge to the unit interval.
c. Take your favorite compact connected Riemannian manifold  $(M,g)$ (say, a round sphere) and a sequence of finite subsets  $X_n\subset  M$ which form $1/n$-nets in $M$ (i.e. for every $n$, each point of $M$ is within distance $1/n$ from some point of $X_n$). Equip $X_n$ with the restriction of  the Riemannian distance function $d_g$ on $M$. Then the sequence $(X_n)$ will converge  to the metric space $(M, d_g)$.
d. Consider the standard (ternary) Cantor set $C\subset {\mathbb R}$, obtained by inductively removing "middle thirds" from the unit interval $I$. Let $C_n$ denote the $n$-th leftover subset of $I$ (the subset left in $I$ after removing middle thirds $n$ times), see here. Equip $C$ and $C_n$ with the restriction of the standard distance on the real line. Then $(C_n)$ will converge to $C$.
