It would seem like there's something circular about pointing to various passages in the Bible to prove that the Bible is the infallible word of God. Take these passages for instance.
John 10:35 "If he called them gods, to whom the word of God came (and the Scripture cannot be broken). . ."
2 Timothy 3:16 "All Scripture is God-breathed and profitable for teaching, for reproof, for correction, for training in righteousness. . ."
2 Peter 1:20-21 "But know this first of all, that no prophecy of Scripture is a matter of one’s own interpretation, for no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God."
Why should we believe that these passages are true? Because they're in the Bible. And why should we think the Bible is accurate? Because these passages say so. So the Bible is true because the Bible says it's true.
Well, that's obviously problematic. It doesn't have to be, though. Consider my imaginary friend, Bob. Bob is a pretty reliable guy. At least 90% of what Bob says is true. The other 10% may or may not be true. Who knows? Well, one day, Bob said, "100% of what I say is true." If we were going on statistics alone, and not considering the particularity of this kind of statement, we'd have to assign the statement a 90% chance of being true. That means there's a 90% chance that Bob is 100% correct in everything he says.
Of course we wouldn't actually do that because there are other reasons to doubt the claim. Bob is human, humans are fallible, therefore, Bob is fallible. Plus, it's kind of a crazy claim anyway. All of those things would lead us to believe that the probability of the claim being true is less than 90%.
But all I'm trying to say is that using Bob to prove Bob is not necessarily circular. It's not in principle circular to rely on Bob's word that everything he says is true. We can rely on his general truthfulness to substantiate his claim.
The Bible differs from Bob, though. Bob is just one person, but the Bible contains the works of multiple people. To say "using the Bible to prove the Bible" is misleading. Consider Bob's imaginary friend, Dan. Dan is a pretty truthful guy, too. At least 90% of what Dan says is accurate. One day, Dan says, "Everything Bob says is true." Well, if Dan and Bob are different people, then there's no circularity in arriving at the conclusion that Bob is 100% accurate in everything he says. Since the Bible has different parts written by different authors, it's not circular reasoning to base the reliability of one author on the word of another author. That's why I say it's misleading to call this an example of "using the Bible to prove the Bible." Using Bob to prove Bob might be circular, but using Dan to prove Bob is not.
Let's say we know that Dan is wrong in 10% of what he says, but we know he's right in 90% of what he says. Not taking other background information into account (like the humanity of Dan and the fallibility of humans), there's a 90% chance that Dan's statement about Bob is true. That means there's a 90% chance that Bob is 100% correct in everything he says. There's no circular reasoning there.
So when Paul say that all scripture is God-breathed, and he's referring to all of the works of the Bible by authors other than himself, then it isn't circular reasoning to appeal to Paul to substantiate the truth of all those other books in the Bible. All we need is some reason to think Paul knows what he's talking about.
Of course if the only reason we accept the reliability of Paul is because Peter says he's reliable (2 Peter 3:15-16), and if the only reason we accept the reliability of Peter is because Paul says he's reliable, then we're using circular reasoning. But as long as we have some reason to think Paul is reliable other than Peter saying so, or vice versa, then our argument is not circular.
And one need not be 100% certain that the Bible is 100% accurate. One can believe in the infallibility of the Bible without claiming that their belief itself is infallible. It is perfectly consistent to say, "I think the Bible is 100% accurate, but I could be wrong about that." So if one is using some evidential or probablistic line of reasoning as the basis for trusting in the Bible's statements about its own authority, and not merely the assumption of infallibility, then one is not reasoning in a circle by concluding that the Bible is infallible from the Bible's own claims of infallibility.
Some people will accuse you of circular reasoning if you use the Bible to substantiate any claim, and not just the claim that the Bible is infallible. For example, let's say it's the claim that Paul met Cephas in Jerusalem. Paul wrote in his letter to the Galatians, "Then three years later I went up to Jerusalem to become acquainted with Cephas, and stayed with him fifteen days" (Galatians 1:18). If you use this verse to substantiate the claim that Paul knew Cephas, the knee jerk reaction of some people is to say, "You're using the Bible to prove the Bible. That's circular reasoning!"
To see how vacuous that retort is, you just have to apply the principle universally. Suppose I cite an article in the news paper to show that a murder took place at Peas Park over the weekend. Would the correct response be, "You're using the newspaper to prove the newspaper"? If so, then none of us could cite a source for anything at all because it would always invite the accusation of circular reasoning. Whatever the source is, one could say, "You're using the Source to prove the Source!" This knee-jerk Bible-to-prove-the-Bible retort is an argument that proves too much since it invalidates any source that you might cite to substantiate any claim.
Suppose we argue for the infallibility of the Bible by a more circuitous route. We begins with the claims of the Bible that (1) Jesus made himself out to be a prophet, messiah, and son of God, (2) Jesus was crucified for his claim to be the messiah, (3) some of his followers claimed to see him alive after his crucifixion, and (4) those same followers proclaimed that Jesus had risen from the dead. From there, we reason that (5) the appearances of Jesus were what caused them to believe he had risen from the dead, so (6) Jesus really did rise from the dead, so (7) Jesus really must've been a prophet, messiah, and son of God, so (8) Jesus was a reliable source. Also, (9) Jesus believed in the absolute authority of the Bible; therefore, (10) the Bible must be absolutely authoritative. Would this be an example of using the Bible to prove the Bible? In a sense, yes, but it's not circular reasoning as long as we don't rely on the divine authority of the Bible to substantiate the premises leading up to the conclusion. It would be circular reasoning if the only reason I believed Jesus made these lofty claims about himself or that he was crucified, etc. was because I already assumed the Bible was absolutely authoritative. But since all of the premises can be substantiated without assuming the divine authority of the Bible, the argument is not circular.
We can all agree that the New Testament is a product of the early church. It is historically value, at the very least, because it tells us something about what early Christians believed and how they lived. Historians do not naively look at primary sources and take their word for whatever they say. They don't treat the Bible that way either. But any ancient writing is historically valuable because it tells you something about the people it came from. There are criteria historians can use to determine if what a source says is true. These criteria do not amount to saying, "It's true because it says it's true." Rather, it's true because the author was in a position to know, or multiple independent people all say the same thing, or it's an embarrassing admission, or whatever. There are multiple reasons to think that what a person says was true other than the mere fact that they said it. If that were not the case, there would never be any point in calling a witness--especially the accused--to the witness stand.
Catholic apologists really do engage in circular reasoning, though. According to them, the only way we can know what books are God-breathed is because the Catholic Church tells us. We need an infallible source, like the Church, to tell us about the other infallible source--the Bible. But how do we know the Catholic Church has this authority? Well, they base that on scriptures such as 1 Timothy 3:15 where it says the Church is the "pillar and support of the truth." But how do we know we should believe 1 Timothy 3:15? Why, because 1 Timothy 3:15 is part of the canon of scripture, which the Church has defined. So we base the authority of the Church on the authority of scripture, and we base the authority of scripture on the authority of the Church. That's circular reasoning.
A Catholic can avoid circular reasoning by grounding the authority of the Church outside of scripture or by grounding the authority of the Bible outside of the Church. At some point, they're going to have to argue like protestants. They're going to have to look at history and engage in historical methods and argue, fallibly, for the authority of one or the other.