It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn ([On discrete harmonic functions.](http://zbmath.org/?q=an:0033.06303) - Proc. Camb. Philos. Soc. , 1949, 45, 194-206). 



> 
> If a bounded function $f : \mathbb Z^2 \rightarrow \mathbb{R}$ satisfies the following condition
>  $$
> \forall (x, y) \in \mathbb{Z}^2,\quad f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$
>  then f is constant function.
> 
> 
> 


([A stronger version of discrete “Liouville's theorem”](https://math.stackexchange.com/questions/51926/a-stronger-version-of-discrete-liouvilles-theorem) also follows from Heilbronn's proof.)


But recently I knew from Alexander Khrabrov that there is an older article of Capoulade with almost the same result ([Sur quelques proprietes des fonctions harmoniques et des fonctions preharmoniques](http://www.emis.de/cgi-bin/jfmen/MATH/JFM/full.html?type=html&an=JFM%2058.0503.03), - Mathematica (Cluj), 6 (1932), 146-151.) 


Unfortunately the last article is not available for me. Does anybody have an access to this paper? Is this really the first proof discrete Liouville's theorem?


(This question is inspired by [Liouville Theorem](https://math.stackexchange.com/questions/59879/liouville-theorem/577055#577055) from [Mathematics](https://math.stackexchange.com/))



The origin of Discrete `Liouville's theorem'
![](https://ilorentz.org/beenakker/MO/Capoulade.png)


[Mathematica (Cluj), **6**, 146-151 (1932)](https://ilorentz.org/beenakker/MO/Capoulade.pdf)



> 
> In a recent article from this journal, Mr. Bouligand has indicated the
>  possibility of a modified proof of a theorem of Mr. Picard: *"a
>  harmonic function that is positive in the whole space is a constant"*.
>  The purpose of this article is to continue along the same lines to
>  prove an analogous result for preharmonic functions.
> 
> 
> 


So this seems to be the required theorem. If help is needed with the French text of the paper, let me know.


