Proof of the existence of the Trinity based on logic or in concrete evidence does not exist. As with many beliefs of religious faiths, the support for basic tenets comes from an individual's belief - the acceptance of ideas or statements that cannot be tangibly shown but are nevertheless accepted as being valid and true.
The concept of the Trinity holds that God is manifested in three forms: God the Father, God the Son, and God the Holy Spirit. While the term "Trinity" is not used in the Bible, evidence of this triune form of God is seen in various places in the New Testament. When Jesus (God the Son) is baptized,
he (Jesus) saw the Spirit of God descending like a dove and lighing on him. And a voice from heaven said, "This is my Son, whom I love; with him I am well pleased. (Matt. 3:16-17)
The voice came from God the Father.
At the end of the Gospel according to Matthew, Jesus commissions His disciples, telling them, "go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit" (Matt. 28:19), specifically naming the three persons of the Triune God.
The Epistles make additional mention of the separate but equal three-in-one nature of God.
The Trinity is not backed up in the Bible. The concept was invented by early Christians and a Christian Pope in the 5th Century AD during the Council of Chalcedon essentially made it blasphemous to believe otherwise.
If you ask me, I think the concept of the Trinity arose in an attempt to explain why, in the Old Testament, God says such things as "Come, let US go down and confuse their languages", and "Come, let Us make man in OUR image and after OUR likeness" (third-person). Christians, who believe that God is the one and only, could not make sense out of why God was speaking as if he was not alone. Therefore, they held that God must exist as a Trinity.
Those stories from the Old Testament however, are glossed over versions of earlier Mesopotamian Polytheistic stories.