  We study pointwise convergence of the fractional Schr\"odinger means along
sequences $t_n$ which converge to zero. Our main result is that bounds on the
maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced
from those on $\sup_{0<t\le 1} |e^{it(-\Delta)^{\alpha/2}} f|$ when $\{t_n\}$
is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results
provide seemingly optimal results in higher dimensions, which extend the recent
work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based
on a localization argument also works for other dispersive equations and
provides alternative proofs of previous results on sequential convergence.
