I've been struggling with the concept of proofs ever since I completed my introductory logic course "Axiomatic Systems". In that course it seemed to be easy. We were pretty much just using various logical methods to prove the properties of real numbers. Now it doesn't seem nearly as simple. I often find myself stumped when my real analysis text claims "by the principle of mathematical induction this is true," offering no evidence whatsoever that the PMI implies anything. This makes me believe that I must not understand basic concepts like that very well. I really need a reference that properly explains these concepts to me, because even looking through my old axiomatic book I see no clear definition of induction. I'm not sure why I expected a good answer from that book. How could I when our assigned text was just over 70 pages for a full semester? (70 pages WITH examples in case you were wondering)
An explanation of mathematical induction would be appreciated, but I really would prefer book recommendations.