Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I⊂P(N) the quotient space l∞(I)/c0(I) is complete. The second main result states that consistently there is an admissible ideal I⊂P(N) such that the sets W(I), of all real sequences with finite I-variation, and c⁎(I), of all restrictively I-convergent sequences, are equal.