So I'm taking my first midterm exam in math analysis soon and I'm confused on how to study for it. I mean in calculus one would be able to just go into the book and take derivatives and integrals until it was second nature, but in analysis, the class is mostly proving theorems and writing proofs of important theorems. I understand that every university and even every class is going to be different but what are some of the essential things that are learned in the first part of an analysis course that would likely be tested on? If anyone is familiar with Charles Pugh's text on Real Analysis the exam covers the whole first chapter and section 1 of chapter 2.
In class we have covered openness and closedness of sets, continuity, the construction of the real number system in the method of Cauchy sequences (professor didnt really touch on them being built on cuts too much, but a little), homeomorphisms, upper and lower bounds, cardinality, etc.
Of course I'm going to study those topics, but is there a good way or a suggested way that tends to help students prepare for those in an exam?
Get the definitions. Try to prove the theorems yourself. If you can't , understand the proof idea and why you missed it. Thinking visually and creating examples may help to solve the problems.