Boundedly complete weak-Cauchy basic sequences in Banach spaces with the PCP (Q2466530)

It is proved that every non-trivial weak-Cauchy sequence in a Banach space with the PCP (the Point of Continuity Property) has a boundedly complete basic subsequence. The following result, due independently to S. Bellenot and C. Finet, is then deduced as a corollary. If a Banach space \(X\) has separable dual and the PCP, then every non-trivial weak-Cauchy sequence in \(X\) has a subsequence spanning an order-one quasi-reflexive space. The main result is proved using the fact that every separable Banach space with the PCP has a boundedly complete skipped-blocking decomposition.