Limit points of subsequences

Abstract Let x be a sequence taking values in a separable metric space and let I be an F σ δ -ideal on the positive integers (in particular, I can be any Erdős–Ulam ideal or any summable ideal). It is shown that the collection of subsequences of x which preserve the set of I -cluster points of x is of second category if and only if the set of I -cluster points of x coincides with the set of ordinary limit points of x ; moreover, in this case, it is comeager. The analogue for I -limit points is provided. As a consequence, the collection of subsequences of x which preserve the set of ordinary limit points is comeager.