(The following is crossposted from [Math.SE](https://math.stackexchange.com/questions/2759503/surjectivity-of-a-map-on-inverse-limits), 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. 



Surjectivity of a map on inverse limits
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*.


