How can you prove the existence of God to an unbeliever, for the purpose of bringing them to salvation?
Saving faith in Jesus Christ comes by faith alone in the word of God:
Rom. 10:17 So faith comes from hearing, and hearing by the word of Christ.
God may choose to work through our intellectual pursuit of Him to reveal Himself to us, but He will only be found if He chooses to reveal Himself to a person. Our goal can't be to “prove” God’s existence to unbelievers. Such a goal is not even obtainable, according to scripture. God cannot be found by men unless, and until, He chooses to reveal Himself. Meanwhile, He is self-evident already:
Rom. 1:20 For since the creation of the world His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse.
Paul says that when a person questions God’s existence, they are denying the obvious.
Nevertheless, the Lord in His grace may chose to explain Himself in His word to such people. Therefore, our response to this question should always be to encourage such a person to read the word of God in the hope they will meet the Creator as He reveals Himself to that person. As Jesus said:
Luke 16:31 “But he said to him, ‘If they do not listen to Moses and the Prophets, they will not be persuaded even if someone rises from the dead.’”
If a person is not persuaded by the word of God, then nothing we say will overcome their objections either. You can read more about the trustworthiness of Scripture here: Explaining Biblical Inerrancy