Borel extractions of converging sequences in compact sets of Borel functions

It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence ( f n ) n ∈ ω in K one can extract a subsequence ( f n k ) k ∈ ω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence.