A bounded
sequence (xn) in a Banach space is called
ϵ -weak Cauchy, for some
ϵ>0, if for all
x⁎∈BX⁎ there exists some
n0∈N such that
|x⁎(xn)−x⁎(xm)|<ϵ for all
n≥n0 and
m≥n0. It is shown that given
ϵ>0 and a bounded
sequence (xn) in a Banach space then either
(xn) admits an
ϵ -weak Cauchy sub
sequence or, for all
δ>0, there exists a sub
sequence (xmn) with the following property. If
I is a finite subset of
N and
ϕ:I→N∖I is any map then
for every
sequence of complex scalars
(λn)n∈I. This provides an alternative proof for Rosenthal's
ℓ1-theorem and strengthens its quantitative version due to Behrends. As a corollary we obtain that for any uniformly bounded
sequence (fn) of complex-valued functions, continuous on the compact Hausdorff space
K and satisfying
limsupn,m→∞|fn(t)−fm(t)|≤ϵ, for some
ϵ>0 and all
t∈K, there exists a sub
sequence (fjn) satisfying
limsupn,m→∞|∫K(fjn−fjm)dμ|≤2ϵ, for every Radon measure
μ on
K.