Core equality of real sequences

Given an ideal $\mathcal {I}$ on $\omega $ and a bounded real sequence $\boldsymbol x$, we denote by $\mathrm {core}_{\boldsymbol x}(\mathcal {I})$ the smallest interval $[a,b]$ such that $\{n \in \omega : x_n \notin [a-\varepsilon ,b+\varepsilon ]\}