A bounded entire function in the complex plane 
is constant. The fundamental theorem
of algebra follows as a simple corollary.
Liouville's Boundedness Theorem
See also
Complex Plane, Entire Function, Fundamental Theorem of AlgebraExplore with Wolfram|Alpha
References
Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, p. 74, 1996.Krantz, S. G. "Entire Functions and Liouville's Theorem." §3.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 31-32, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 381-382, 1953.Referenced on Wolfram|Alpha
Liouville's Boundedness TheoremCite this as:
Weisstein, Eric W. "Liouville's Boundedness Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LiouvillesBoundednessTheorem.html