Is The Bible truly infallible? Is it inerrant without divine inspiration?

My question comes in two parts, the second being a direct result of the first. **Is the Bible infallible?** I have often heard that it is, but the most conclusive proof that I have heard essentially states that, since it is inerrant, it must be infallible. This makes no sense as it is inerrancy is necessary due to infallibility, not the other way around. When bringing into question the infallibility of The Bible. I am not asking about The Canon, or the books selected, but rather the actual content of scripture. **If The Bible is not infallible, then on what grounds do we say it is inerrant?** Assuming that The Bible cannot be proven infallible, can we at least say it is inerrant? The Bible certainly is proven legitimate with historical evidence and its teachings are proven true in practice. This points to the conclusion that it is inerrant, but does it conclusively and certainly prove its inerrancy? Is there any way to show that, without a doubt, The Bible is inerrant? **EDIT:** To clarify what I’m asking further, I am defining - **Infallible**: Without ability to err - **Inerrant**: Without error - **The Bible**: The actual, original content of widely accepted Scripture and its meaning My first question can also be examined as a question of Divine Inspiration. Is every word a product of God’s Will?