Liouville's Theorem (idea)

Theorem.

Any bounded (holomorphic) entire function is constant.

In fact, considerably more than this is known: the real and imaginary parts of a holomorphic function are harmonic functions; not only do no bounded harmonic functions exist in R^n, but in fact minimal rates of growth may be established.

Actually, proving a discrete version of this theorem is rather more difficult than the continuous versions -- see Z^n admits no bounded harmonic function for more details.