To say that the scriptures are "inerrant" literally means that they are "without error." That is, that there is not one single error in spelling, grammar, fact, etc. any where in all of scripture. Jesus said in Matthew 5:18 "For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished." Based on Jesus' declaration, we conclude that there cannot be one single error in recording even the smallest letter or stroke anywhere in the scriptures. However, that is only true of the original documents (or autographs) as they were written by the various authors who were moved by the Spirit of God (2 Peter 1:20-21). In this process God somehow had His perfect word "breathed out" or "inspired" (2 Tim 3:16) and it was without error. We believe this by faith because of numerous statements in the Bible by God about the perfect nature of His word (e.g. Psalm 19). There is no external evidence that can prove the inerrancy of those original autographs.
After those writings were recorded without error, they were carefully copied by thousands of scribes or copyists over many centuries and those copyists did make some errors in copying. We call those copying errors "variants" in the text. Because we have so many copies (e.g. 25,000 manuscript copies of the New Testament), we are able to identify where most of the errors are and we have been able to produce a text of the Bible in its original languages for which we are certain that we have been able to recover the exact wording and spelling for almost 95% of the text. So, although the Greek, Hebrew and Aramaic texts of scripture that we have today do contain some errors (mostly in spelling, word order, grammar, etc.), the original texts did not.
Thus, the Chicago Statement on Inerrancy in Article 10 states:
We affirm that inspiration, strictly speaking, applies only to the autographic text of Scripture, which in the providence of God can be ascertained from available manuscripts with great accuracy. We further affirm that copies and translations of Scripture are the Word of God to the extent that they faithfully represent the original.
("Inspiration" implies "inerrancy.")
Making an error is one thing; being wrong is another. The word "infallibility" literally means "not able to fail." It simply means that the scriptures are never wrong. They have never failed to be true or failed to come true. Every word of God is tested and proves true (Proverbs 30:5). God's word is truth (John 17:17). This is the case for every single word that the Bible records throughout its entirety and on every single subject that it addresses including matters of science, geography, history, relationships, church doctrine, the future, eternity, etc. (in recognition of various figures of speech). Thus, you can rely on it to be true and never fail. Because God's word, the Bible, is infallible, it is trustworthy.
Contrary to the lack of external proofs for inerrancy, there is much external evidence that can prove the infallibility of the scriptures. Archeology has demonstrated the infallability of the word of God time and time again in recent years. Many times, names and places recorded in the Bible, although questioned by some scholars in the past, have been proven to be accurate. The exact fulfillment of thousands of prophecies clearly demonstrates that God's word is true.