Limit points of subsequences

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.