(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 did not manage to come up with a proof), 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.

(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.