I have a certain sequence $x_n$ in a complete and **bounded** metric space and I would like to prove that it has a convergent subnet (not necessarily subsequence). The best that I was able to do, until now, is to prove that $x_n$ verifies the following property:

for all $K\in\mathbb N$ and for all $\epsilon>0$, there is $n_{\epsilon,K}$ such that $d(x_n,x_m)<\varepsilon$, for all $n,m\geq n_{\epsilon,K}$ such that $|m-n|\leq K$.

**Question:** Does this sequence have necessarily a convergent subnet.

I really would like this is true, even if I suspect it is not. Nevertheless, I am not able to find a counterexample. I have also tried to prove that this is indeed true: since $X$ is a complete and bounded metric space, it is paracompact and by a theorem of Howes (th. 6.20 of his book "Modern Analysis and Topology") it would suffice to prove that $x_n$ is almost Cauchy. But it seems to me that my hypotheses do not imply that $x_n$ is almost Cauchy, since none of my sets where $x_n$ is Cauchy is cofinal.

Thanks in advance for any help,

Valerio