I am currently taking analytical mechanics,
My professor directed us in the lecture to Hand and Finch, problem 6.9, to prove ourselves (as he didn't have time) that the equation $$[Q,P]_{Q,P} = [Q(q,p),P(q,p)]_{q,p} = 1\tag{6.39}$$ is both necessary and sufficient condition for the transformation between $q, p$, and $Q, P$ to be canonical (the exact phrasing of the question appears in the attached picture below).
I'm having trouble figuring out where to begin with this proof. Could someone provide a proof?
Please keep in mind that we haven't covered symplectic notation in class, so explanations without this concept would be greatly appreciated.
