Cauchy subsequences in "Hausdorff Cauchy sets"

Question

This is a follow-up to an older question.

Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of bounded 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).

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?

Answer 1

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. So this fails even if the sequence $\langle A_n\rangle$ is constant.