A (non-trivial) ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is a p-point if and only if every bounded sequence $(a_n)$ of real numbers contains a convergent subsequence $(a_{n_k})$ such that $\{n_k\}\in \mathcal{U}$.
See [ISRAEL JOURNAL OF MATHEMATICS 122 (2001), 189-206] for an application of these ultrafilters to obtain an ultrapower version of the subsequence splitting property for $L_1(\mu)$, some discussions related with the question, and proper references.