Let $\omega$ be a nonprincipal ultrafilter 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 $J\in\omega$?