Tauberian theorems for ordinary convergence

We show that a real sequence x is convergent if and only there exist regular matrix A an Fσδ-ideal I on N such the set of subsequences y for which Ay I-convergent second Baire category. This includes cases where ideal asymptotic density zero sets, Banach finite sets. The latter recovers old result given by Keogh Petersen in (1958) [26]. Our proofs are different nature rely recent results context I-Baire classes filter games. As application, we obtain stronger version classical Steinhaus' theorem: each A, exists {0,1}-valued Ax not statistically convergent.