Generalization of Riemann's Rearrangement Theorem

Abstract

In this paper, we will examine Riemann's rearrangement theorem for sequences of real numbers. Continuing on, we will briefly discuss previous generalizations of the theorem to the n dimensional case. Because of the difficulty of that proof, we present a more approachable proof for the case of complex numbers. Our proof is not significantly more difficuult than the proof for real numbers. Furthermore, key aspects of the proof can be applied to the n dimensional case. From there, we are able to prove the n dimensional case using elementary techniques. We conclude by considering the possibility of an infinite dimensional generalization through sequence or function spaces.