Is there a sequence with no convergent subsequence but small distances between terms?

Question

Is there a metric space $(X, d)$ and a sequence $(x_{n})$ in $X$ that has no convergent subsequence but infimum of the set of distance between elements of the set is zero?

To be more precise, $\inf\{d(x_{m},x_{n}) | n,m\in N,\,m\ne n\} = 0$

Answer 1

Let $X$ be the real subspace $\{\log n: n\in \mathbb N\}$ with the usual metric $d(x,y)=|x-y|.$ Since $0<\log (n+1)-\log n=\log (1+1/n)<1/n$ we have $\inf \{d(x,y):x\ne y\}=0$. But $X$ is a discrete space.

Or let $(x_n)_{n\in \mathbb N}$ be any other strictly increasing real sequence with $\lim_{n\to \infty}(x_{n+1}-x_n)=0$ and $\lim_{n\to \infty}x_n=\infty.$ And let $X=\{x_n:n\in \mathbb N\}$ with $d(x,y)=|x-y|.$

Answer 2

Let $\Bbb Q$ with the standard metric $d(x,y)=|x-y|$ and let $\displaystyle (s_n)=\sum_{k=1}^n \frac{1}{n^2}$. Then $s_n \to \dfrac{\pi^2}{6} \notin \Bbb Q$ and therefore:

(i) $(s_n)$ is Cauchy implies $\inf\{d(s_m,s_n): m, n\in \Bbb N\}=0$

(ii) every subsequence of $(s_n)$ converges to $\frac{\pi^2}{6} \notin \Bbb Q$.