We can describe the Trinity, and defend the Biblical teachings about the Trinity against misunderstandings and misrepresentations. But I'm not sure you can "prove" the Trinity, in the normal sense of what we mean by that word. It's hard enough to describe something that has no parallel in the creation. The Trinity is sui generis -- one of a kind. So I believe in the Trinity because of the nature and teachings of Jesus Christ. I take it as an article of faith revealed to us in Scripture, not because I can naturally reason upward to the Trinity from things I see in nature.
Jack is the teaching elder of Ironworks Pike Community Church. He is a graduate of Columbia Biblical Seminary http://www.ciu.edu.