Wow. There is not a simple answer to that question.
But I will see if I can help a little bit.
First, it is impossible to prove inspiration from the Bible itself. There would have to be somewhere in Scripture where there was a list of al the books which are inspired by God. There is no such list. Even if Scripture did testify to itself, it would still not be valid. That would be circular logic. “Is the Bible inspired?” “Yes.” “How do you know?” “The Bible says so.” I could say I am a mermaid. That does not make it true.
What you must do is find some proof outside of the Scriptures. Ultimately, the only reason anybody knows that the Bible is the inspired, infallible word of God is because the Catholic Church says it is. But why should your friend believe the Church? Because it is the Church Jesus established. But why should your friend believe in Jesus? For that we have to look at scripture. Pulling your hair out yet? You see, there is a lot more to your question than you may think. It certainly is not a simple question.
I will try to briefly summarize, based on the assumption your friend believes in God. If he doesn’t, you have even more work to do.
First, since Christianity is based on the person of Christ, you have to establish that the extant histories of Christ (the Gospels) are reliable historical documents. This is different than proving inspiration. This is treating them as any other historical work of Aristotle, Cicero, or Thomas Jefferson.
Are the Gospels, as we have them now, the same as they were two thousand years ago? Or have they been corrupted? Based on the fact that we have more than two thousand complete and partial manuscripts from the very early Church, going to the mid second century, we can compare what we use now and see that they are essentially the same.
Who actually wrote the Gospels? Based on extensive external and internal evidence, we can say with certainty (more certainty than most other historical documents) that they were written by two apostles who followed Jesus everywhere, and two men who followed the apostles everywhere. Were they in a position to actually know the historicity of the events the recorded? You bet.
Are these true histories? Yes. These were all written and widely disseminated while there were many people still alive who were around when these events took place. There would certainly be some record of disagreement if anything proved false.
So the gospels are accurate historical documents. What is next is to look at what they say. Jesus says He is God. Is He a lunatic? How can we know He tells the truth? When asked, He pointed to His miracles. He healed people, changed water to wine, multiplied bread, and many more. Since these are true accounts of what He did, we must conclude, based on the impossiblity of a human performing these actions, that only God could do them. We also have His prophecies, which have been fulfilled. He prophecied about Himself, Jerusalem, and how His disciples would be treated. He was very specific in all these prophechies, which means He didn’t just get lucky when they came to pass. He actually knew what was going to happen, in a way only God could know. Based on these proofs, Jesus must be who He said He is.
Why did God come to earth? Two reasons. First, to redeem mankind, save him from his sins, and reconcile him to God. Second, to teach us what we need to do to attain Heaven.
I hope that helps a little. What remains to be proven is that Jesus established His Church as His legitimate teaching authority on earth, and that we should follow Her and believe Her teachings, which include the fact that Scripture is inspired. I can go into that more later, but I feel I have already gone bit long on this.
A great book you might want to check out is called “College Apologetics” by Fr. Anthony Alexander. I believe it is published by Tan Books. You may be able to get it from Catholic Answers.
God bless.
