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 [M. Gonzáles, A. Martinéz-Abejón: *Ultrapowers of $L_1(\mu)$ and the subsequence splitting principle*, Israel Journal of Mathematics 122 (2001), 189-206, [DOI: 10.1007/BF02809899](http://dx.doi.org/10.1007/BF02809899)] 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.