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In Mathew 28:18-19 Jesus said, "All authority in heaven and on earth has been given to me. Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit."
At the time of His ascension Jesus said to His disciples, "But you will receive power when the Holy Spirit comes to you," Acts 1:8.
John 16:5-16 makes it clear that the Holy Spirit is not to be found in Heaven, but with us and in us. In John 16:7 Jesus makes this very clear when He says, "It is for your good that I am going away. Unless I go away, the Counselor will not come to you; but if I go, I will send him to you."
John 14:15-31 is also very clear about the existence of the Holy Spirit; specifically John 14:16 where Jesus says, "And I will ask the Father, and he will give you another Counselor to be with you forever." Also, John 14:26 where Jesus says, "But the Counselor, the Holy Spirit whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you."
Jesus' words are enough to convince me that the Holy Spirit does, in fact, exist and is a vital part of the Trinity. He does not exist in heaven; but right here with you and me guiding everyone who truly loves Jesus and believes that He is their Savior.
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If an apple can be 3 in 1, is it so difficult to imagine that God, with whom all things are possible, could also be 3 in 1. Yes, the Trinity does exist and Christians worship one God consisting of three inseparable persons.