TITLE: Surjectivity of a map on inverse limits
QUESTION [7 upvotes]: (The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, Lemme 1):

Lemma. Let $B$ be a ring, $(I, \leq)$ be a directed poset, and $(M_i, f_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, g_{ji}:N_j\rightarrow N_i)_{j\geq i}$ two inverse systems of $B$-modules. Let $(h_i: M_i\rightarrow N_i)_i$ be a morphism of the inverse systems, and assume that all the maps $h_i$ are surjective with an Artinian kernel. Then the limit map
  $$\varprojlim_i h_i: \varprojlim_i M_i\rightarrow \varprojlim_i N_i$$ is surjective.

I am looking either for a proof (I haven't manage to come up with one so far), or for a reference to a proof - Gabriel refers to "Bourbaki, Topologie, I Appendice, $3^{\text{e}}$ éd.", which is a reference I cannot find anywhere.
I do not want to assume that e.g. $I$ is countable - the reason is that I need this lemma, similarly as Gabriel in his thesis, to establish some properties of pseudo-compact modules over a pseudo-compact ring (namely that quotient of a pseudo-compact module by a closed submodule is pseudo-compact, and exactness of inverse limits). For this reason, as far as I can tell, I cannot use countable index sets (i.e. countable bases of neighbourhoods) in general.
Thank you in advance for any help.

REPLY [4 votes]: Your question can be interpreted as the vanishing of the first derived projective limit functor ${\lim\limits_{\leftarrow}}^{(1)} \mathcal M$ for the projective spectrum of the kernels. If these kernels are Artenian even all derivatives vanish. This is shown in Corollary 7.2 of C.U. Jensen's Les Foncteurs Dérivés de lim  et leurs Applications en Théorie des Modules, Springer Lecture Notes 254 (1972). He also refers to his article On the vanishing of ${\lim\limits_{\leftarrow}}^{(i)}$, J. Algebra 15 (1970), 151-166 (but I did not have a look at it).
It seems to me that Jensen does not claim for originality as he writes "Il est bien connu...". However, he does not give an explicit reference. A main step in his proof is a result in Bourbaki's Topologie Générale.