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 contex...