Reddit reviews Advanced Calculus (Pure and Applied Undergraduate Texts: The Sally Series)

It's a rigorous upper-level math course focused on proving the fundamental theorems of calculus using basic logic and building up from set theory. Can confirm, was my lowest grade the semester I took it since unless you're Gauss you can't do the homework or tests without going to office hours.

Example question: "Prove this limit of this fucked up fractal looking puzzle piece bullshit function converges to 2" and you'd have to do a proof by analyzing the function's continuity, using some limit properties (which we defined using epsilon-delta, as well as a not so constructive definition using convergent sums), and then maybe using the modular properties of the fractal curve to simplify it into something that you can actually compute using basic algebra (which you also need to prove, obviously).

It's pretty cool though. Nice to see that math is super consistent and builds on its own concepts just starting with a few naive assumptions about what works (axioms). We literally went from set theory, and a single element "1" and another element "0" to the entire set of real numbers by defining "2" in terms of "1", defining "1/2" in terms of 1 and 2, and filling in the irrational numbers after proving the rationals are dense (which has a specific mathematical meaning) but not dense enough to be consistent with a few properties, so we need the irrational numbers to get the math we're used to.

Do I recommend taking it? Hell no! Read this reddit post a few times until it makes even less sense than the first time you read it and you have the gist of the class. Or read this unhelpful textbook: https://www.amazon.com/Advanced-Calculus-Applied-Undergraduate-Texts/dp/0821847910

For your 14-month course:

A good book on probability theory and advanced statistics: I'm fond of Freund's Mathematical Statistics with Applications

Linear algebra: Lay's "Linear Algebra and its Applications" is all right for a first course.

The analysis course sounds a bit like what I had out of my Sally Series Advanced Calculus text.

I would grab the 410 textbook and start reading chapter 1 (first 2 sections) and chapter 2 (first 4 sections). Chapter 2, especially section 2.1, should be a pretty good indicator of your ability to succeed in 410. If you can follow the proofs and reproduce them on your own, you'll probably be fine without 310.