A subsequence principle is obtained, characterizing Banach spaces containing
c
0
{c_0}
, in the spirit of the author’s 1974 characterization of Banach spaces containing
ℓ
1
{\ell ^1}
. Definition. A sequence
(
b
j
)
({b_j})
in a Banach space is called strongly summing (s.s.) if
(
b
j
)
({b_j})
is a weak-Cauchy basic sequence so that whenever scalars
(
c
j
)
({c_j})
satisfy
su
p
n
∥
Σ
j
=
1
n
c
j
b
j
∥>
∞
{\text {su}}{{\text {p}}_n}\parallel \Sigma _{j = 1}^n{c_j}{b_j}\parallel > \infty
, then
Σ
c
j
\Sigma {c_j}
converges. A simple permanence property: if
(
b
j
)
({b_j})
is an (s.s.) basis for a Banach space
B
B
and
(
b
j
∗
)
(b_j^ * )
are its biorthogonal functionals in
B
∗
{B^ * }
, then
(
Σ
j
=
1
n
b
j
∗
)
n
=
1
∞
(\Sigma _{j = 1}^nb_j^ * )_{n = 1}^\infty
is a non-trivial weak-Cauchy sequence in
B
∗
{B^ * }
; hence
B
∗
{B^ * }
fails to be weakly sequentially complete. (A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.) Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either an (s.s.) subsequence or a convex block basis equivalent to the summing basis. Remark. The two alternatives of the theorem are easily seen to be mutually exclusive. Corollary 1. A Banach space
B
B
contains no isomorph of
c
0
{c_0}
if and only if every non-trivial weak-Cauchy sequence in
B
B
has an (s.s.) subsequence. Combining the
c
0
{c_0}
- and
ℓ
1
{\ell ^1}
-Theorems, we obtain Corollary 2. If
B
B
is a non-reflexive Banach space such that
X
∗
{X^ * }
is weakly sequentially complete for all linear subspaces
X
X
of
B
B
, then
c
0
{c_0}
embeds in
B
B
; in fact,
B
B
has property
(
u
)
(u)
. The proof of the theorem involves a careful study of differences of bounded semi-continuous functions. The results of this study may be of independent interest.