We continue the previous talk and focus on $C(K)$ spaces which
satisfy the Grothendieck property. In the beginning we recall some
notions related to sequential convergence in Banach spaces, namely
weakly Cauchy sequences, and convex block subsequences (which can be
understood as a generalization of the notion of a subsequence). Finally,
we will use these notions to prove that consistently every non-reflexive
Grothendieck Banach space contains $l_1(2^\omega)$, and thus admits $l_\infty$ as a
quotient. This, together with the result that non-Grothendieck $C(K)$
spaces contain a complemented copy of $c_0$, will give us a dichotomy
result about $C(K)$ spaces.