Rephrase: Does the Bible claim itself to be inerrant? Before I go on, I do realize that biblical claims for the inerrancy of the Bible is in fact self-satisfying and therefore questionable (at least in a court room), but bear with me.

The old testament scripture is verified by Jesus's obvious regard for it as He quotes it all the time (See Matt. 19:4, Mark 10:6, Heb 4:4, Acts 7:3, the list goes on...). 

I could see the words of Jesus while he was on earth (mainly the gospels) verified as Jesus allowed the Holy Spirit to help the apostles remember what they had heard (John 14:26).

But what about the epistles and apocalyptic books? We can obviously assume that the apostles writing the books knew mostly what they were talking about as they spent so much time with Jesus, but there's a pretty big leap from that to infallible. We also don't know who wrote some books (Hebrews for example), how do we know these are the words of God?

Worth mentioning is 2 Timothy 3:16:


  All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness.


But how do we know what "Scripture" Timothy is referring to here? And as for the "canonization" of the books, are we just trusting that God safeguarded the accuracy of the Bible by guiding the decisions of those involved in the Council of Trent and Synod of Hippo, for example?

If there is no such claim, why did some Christian denominations start believing it was inerrant?

Thanks and God Bless!

CLARIFICATION: I know any answer supporting the inerrancy of the Bible can't, by nature, be a perfect proof because of the recursiveness, but I think it worthwhile to find out if the Bible claims itself to be inerrant or if people just assumed it was later.
See "The Case for Christ" by Lee Stroebel, chapter 3, section "A High Degree of Unanimity" (I can't post it here, because "No part of this publication may be reproduced" etc.) It talks about the rigorous process used to select books as part of the canon. They had to be written by apostles or eyewitnesses to them, they had to conform to existing Christian doctrine, and they had to be accepted by the community at large. Even today there are some eastern churches that don't accept the book of Revelation.