In this paper, we are interested in finding sufficient conditions on a Borel set X lying either inside a bounded domain D ⊂ Cn or in the boundary ∂D so that if {rm}m≥1 is a sequence of rational functions and { fm}m≥1 is a sequence of bounded holomorphic functions on D with { fm − rm}m≥1 is convergent fast enough to 0 in some sense on X then the convergence occurs on the whole domain D. The main result is strongly inspired by Theorem 3.6 in [3] whether the { fm} is a constant sequence