On Proofs (Week 5)

The structure of a proof in this course actually makes a lot of sense. However, some of the rough work gets confusing at times, especially when the only viable option seems to be to use a "definition" of something. Although the definitions of most things makes sense after the professor writes it down, I believe that I would not have been able to conceive the definition as a way to proceed in a proof. What I mean is that I would not be sure what I can use and when I would use it. In the professor's examples, he would of course have already known before hand what he could use but if I were given a new proof of something I am completely unfamiliar with, I do not believe that I could proceed far into that proof before getting stuck.
Compared to delta-epsilon proofs of limit or mathematical proof by induction (from MAT137), I find plain proofs (the ones that just says "prove") to be the most difficult to approach. Mainly it is because it seems like things are somethings pulled out of thing air and used to proceed in the proof steps. I understand that we can assume things to be true and then we would indent and thus enter a world where said thing is true. I understand that we need to end by concluding the statement is true or that its false by contradiction. However, most of the initial steps of the "prove something" parts seems like pure magic. Perhaps I do not know enough laws or theorems to approach it the same way as the professor. I guess my major focus now in this course is to learn why something can be used in a proof.