Bounded harmonic function is constant [closed]

E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995

9 lines long. Not the shortest paper ever, but maximizes importance/length

...

http://www.jstor.org/stable/2034412

http://en.wikipedia.org/wiki/Harmonic_function#Liouville.27s_theorem

Edward Nelson's paper is freely and legally available here:

$\bullet\ $ pdf file,

$\bullet\ $ html page.

If $u$ is a harmonic function then there exists a conjugate function $v$ and an analytic function $f=u+iv$. Thus $\exp(f)$ is bounded, applying the Liouville's Theorem shows that $\exp(f)$ is constant.It's easy to prove that $f$ is constant, as well as $u$.