Everyone knows the usual Cauchy criterion for metric spaces: $(x_n)_n$ is Cauchy if $$\forall \epsilon>0,\,\exists N \text{ such that }\forall n,m\geq N, d(x_n,x_m)\leq \epsilon.$$
Instead, I am interested in sequences $(x_n)_n$ that satisfy the following condition (C): for every sequences $\phi,\psi:\mathbb{N}\to\mathbb{N}$ satisfying $\lim_n \phi(n)=\lim_n\psi (n)=+\infty$, we have $$\lim_n d(x_{\phi(n)},x_{\psi(n)})=0.$$
Obviously, every Cauchy sequence satisfies (C). It is very natural to expect that (C) is actually equivalent to Cauchy criterion, but I haven't succeeded in proving it (except in some easy cases that I will mention bellow). My question: is this equivalence true or is there a wild counterexample?
For real sequences the equivalence holds, because we can just take $\phi$ and $\psi$ that realise $\liminf x_n$ and $\limsup x_n$. Also, it holds when the metric space is compact.