  Let $x$ be a sequence taking values in a separable metric space and
$\mathcal{I}$ be a generalized density ideal or an $F_\sigma$-ideal on the
positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam
ideal or any summable ideal). It is shown that the collection of subsequences
of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$
[respectively, $\mathcal{I}$-limit points] is of second category if and only if
the set of $\mathcal{I}$-cluster points of $x$ [resp., $\mathcal{I}$-limit
points] coincides with the set of ordinary limit points of $x$; moreover, in
this case, it is comeager. In particular, it follows that the collection of
subsequences of $x$ which preserve the set of ordinary limit points of $x$ is
comeager.
