TITLE: Nelson's proof of Liouville's theorem
QUESTION [29 upvotes]: The paper "A proof of Liouville's theorem" 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?

REPLY [10 votes]: This proof was new to me when I read it:-) The standard proof, which I teach, and which is given in most books uses Harnack's inequality, which follows from Poisson's formula for the ball, or Poisson's formula directly.
If I were the editor or a referee, I would accept this paper.