This is a follow-up to [an older question](https://mathoverflow.net/questions/273614/hausdorff-distance-and-cauchy-sequences).


Let $(X,d)$ be a metric space and let $(A\_n)\_{n\in\mathbb{N}}\subseteq X$ be a sequence of [bounded](https://en.wikipedia.org/wiki/Bounded_set#Metric_space) non-empty subsets of $X$ such that for all $\varepsilon > 0$ there is $N\in\mathbb{N}$ such that for all $m,n\geq N$ we have $d\_H(A\_m,A\_n)<\varepsilon$ (where $d\_H(\cdot,\cdot)$ denotes the [Hausdorff distance](https://en.wikipedia.org/wiki/Hausdorff_distance)).


For every $n\in\mathbb{N}$ pick $x\_n\in A\_n$. Does the sequence $(x\_n)\_{n\in\mathbb{N}}$ contain a Cauchy subsequence?



Cauchy subsequences in "Hausdorff Cauchy sets"
No. The unit sphere in any infinite dimensional normed space contains a sequence of elements that all have distance $1/2$ between them by [Riesz's lemma](https://en.wikipedia.org/wiki/Riesz%27s_lemma). So this fails even if the sequence $\langle A\_n\rangle$ is constant.


