Let $\omega$ be a [nonprincipal ultrafilter][1] on $\mathbb N$.

A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers. 
Namely, there is unique real value $x_\omega$ such that  
$$\{\,n\in\mathbb N\mid  |x_\omega-x_n|<\varepsilon\,\}\in \omega$$
for any $\varepsilon>0$.

Clearly $x_\omega$ is a partial limit of $x_n$ [i.e., $x_\omega$ is a limit of a subsequence of $(x_n)$].

>**Question.** Is it always possible to choose subsequence $(x_n)$, $n\in J$
converging to  $x_\omega$ and such that $J\in\omega$?

  [1]: http://en.wikipedia.org/wiki/Ultrafilter