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We characterize Banach lattices in which every bounded sequence contains a subsequence which either converges in norm or satisfies a lower 2estimate. As a consequence we obtain, for the class of all Banach lattices, a positive answer to a question of D.J. Aldous and D.H. Fremlin whether a Banach space of cotype 2 satisfies the above-mentioned property. D. J. Aldous and D. H. Fremlin proved that every AL-space E has the following property (cf. [1, Theorem 6]): (AF) Every bounded sequence (x,) in E has a subsequence which either converges in norm or satisfies a lower 2-estimate. In this note we characterize the Banach lattices satisfying (AF). As a consequence we obtain that every Banach lattice of cotype 2 has property (AF) which solves a problem of D. J. Aldous and D. H. Fremlin (cf. [1, ?9]) for the class of all Banach lattices. Our notation is standard and follows [3]. Let us recall the following definitions. A sequence (x,) in a Banach space E is called semi-normalized if it is bounded and infn I XnII > 0. A sequence (xn) in E satisfies a lower 2-estimate if there exists c > 0 such that for every n E N and every choice of scalars al , ... , ac E R, we have c(Z=1 ak2)/2 < Z1 cEnakxkH . A sequence (xn) in a Banach lattice E is called disjoint if inf(lxnl, IXm) = 0 for all n, m E N with n : m. We begin with a lemma which occurs implicitly in the proof of Theorem 6 in [1]. Lemma. Let E be a weakly sequentially complete Banach space. Suppose every semi-normalized weakly null sequence in E contains a subsequence satisfying a lower 2-estimate. Then E has property (AF) . Proof. Let (xn) be a bounded sequence in E such that no subsequence is norm convergent. If (xn) does not have a weak Cauchy subsequence, then, by Rosenthal's 11-theorem (cf. [2, 2.e.5]), there is a subsequence equivalent to the Received by the editors May 26, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20, 46B30.

Lower 2-estimates for sequences in Banach lattices