[buckykatt]

My sense is that on TM, "real analysis" is mostly a shorter way of saying "an advanced course where the students are expected to produce original, nontrivial proofs on a regular basis".

So, for example, my school--like many US schools--has a course called Discrete Math (or Discrete Structures--I can never remember) that serves as the "intro to proofs" course. (Math and computer science majors typically take it their first year.) It can include a lot of different things, e.g. probability, number theory, graph theory, etc., but it's sure to include relations/functions, set theory, and plenty of practice writing proofs of all kinds. Personally, I think its main function is to weed out the Java code jockeys from the mathematicians, but I'd think that even this course would send a useful signal to an adcom that you aren't going to show up to math camp asking why the A's are upside down and the E's backward.

Our real analysis sequence is two semesters, and is required of all math majors. (Although we cover limits, power series, and such in the regular calculus sequence, the level of the students in those classes is very mixed, and the emphasis is on applications, so there's no expectation that students be able to reproduce the proofs that are presented in lectures.) As a rough guide, I'd say that it picks up somewhere in the middle of the first course you listed above and ends somewhere in the middle of the third. (We don't get into measure theory and the Fourier series gets no more than the occasional mention.) Many students at my school are education majors, so the focus is on the conceptual underpinnings of the calculus, but we managed to cover pretty much everything I know I'll need for the first year of a PhD program. I wish we had done more, but IMHO this course should satisfy any adcom that I'm ready to tackle PhD econ.

How do adcoms know what the content of a course is? Well, they probably don't, exactly. But I doubt that they care, exactly, so long as they're convinced that you really can read and write substantial proofs and know why you want your function defined on a compact set.