The paper "[A proof of Liouville's theorem](http://www.ams.org/journals/proc/1961-012-06/S0002-9939-1961-0259149-4/)" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in $\mathbb{R}^n$ is a constant.


I presume there must be a story behind. First, it is hard to imagine that this proof was unknown before 1961. Second, even if this is the case, it doesn't feel usual, for the author, to submit such a paper and, for the editor, to accept it.


So, can anyone tell that story? Or, to make the question precise:


1. are there any earlier references for this proof?
2. what was/were the standard proof(s) before 1961?
3. by a very similar reasoning, one obtains $̣\|\nabla h\|\_{\infty,\Omega}\leq C(\Omega,\Omega')\|h\|\_{\infty, \Omega'}$ for $\Omega\subset\Omega'$. Was *that* argument also unknown until 1961?



Nelson's proof of Liouville's theorem
This doesn't answer any of the three specific questions asked, but addresses an implicit question: "Why did the editor accept it?"


In 1961, the Proceedings of the AMS established a section called "Mathematical Pearls" devoted to, I quote:



> 
> The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which normally there is no other outlet. 
> 
> 
> 


In the [issue](http://www.ams.org/journals/proc/1961-012-06/) in which Nelson's proof appears, that section starts on page 991 and continues to the end, including 7 papers in all, none of which exceeds 2 pages. In a different issue you can find [this paper](http://www.ams.org/journals/proc/1961-012-04/S0002-9939-1961-0152998-2/S0002-9939-1961-0152998-2.pdf) which also contains the quoted disclaimer above. 


