Invariance of Ideal Limit Points

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an $F_\sigma$-set [resp., a closet set]. Let us assume that $X$ is also separable and the ideal $\mathcal{I}$ satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of $(x_n)$ is equal to the set of ideal limit points of almost all its subsequences.