He begins by picking up discrete calculus, including proof by induction
, and moves to selected area computation (pi, anyone?), limit's and Taylor's theorem, including series representations and Taylor polynomials, infinite series, including both the positive and the general, beginning logic, including propositional logic, predicates and quantifiers, and proofs, real numbers, functions such as derivatives and a substantial pair of chapters on integrals.
I illustrate a student in real analysis who followed this learning progression to learn the concept of proof by induction
. Finally, I note that many students who view proof as a process fail to ever view proof as an argument and I discuss the consequences of these students' narrow view of proof.