There are many systems that have been researched in the past on automating the process of theorem proving in first-order logic. This research explores one of these systems, the tableau method. A point of interest within the tableau method is whether or not the method is sound and complete. This research was done in tandem with a computer implementation of the tableau method written in Haskell. The basic design of the implementation was to construct a fair rule for tableau expansion and expand the tableau until it was found to be closed, open, or infinite, thereby proving or disproving of the theorem. The results of this research find that the tableau method of theorem proving is sound and complete. However, when implemented through software it is an incomplete system due to restrictions in computer processing capabilities and data storage.