I am wondering whether the following result holds: Suppose $(P_n)$ is a sequence of probability measures on a complete separable metric space and every subsequence of $P_n$ has a further subsequence which is tight. Then $(P_n)$ is tight.
In the real setting (or even $\mathbb{R}^d$), the result is true: see Prokhorov's Theorem-Prove if tight subsubsequence, then tight sequence..
I have tried to mimic the proof given, but I am running into some difficulties. In fact, I am not sure the given solution is fully incorrect (the inequalities seem to be the wrong way round).
Anyways, I believe the proof should be quite similar with some modifications, by constructing a nested increasing sequence of compact sets (similar to the $[-j,j]$ in the real setting. Could anyone verify this? Thanks