So you'd like to learn pure mathematics. Where should you start? Real analysis? Topology? Number Theory? You can find introductory, undergraduate, preliminary, and even "gentle" textbooks written on any topic you can think of, and it's not obvious that one subject "comes before" another the same way that algebra comes before pre-calculus. Likewise, plenty of syllabi from august universities claim that their courses are "students' first introduction to rigorous mathematics," only for others to puzzlingly claim "no prerequisites." What's the deal, then? Which subject should you really study first?
The first calculus section on sequences is here. Sequences aren't entirely the most interesting subject in calculus because they're little more than functions that only take integers. The more interesting kind of sequences are sequences whose terms references previous terms. These are called recurrence relations and are studied in more detail in differential equations. Anywho, for now, vanilla calculus sequences are now available.
What's the best subject to start learning proof-based mathematics? With a dizzying array of subjects to pick from, it's hard to tell which you're ready for and which are specialized subfields of specialized subfields.
Most college curricula offer real analysis, complex analysis, number theory, abstract algebra, and probability theory as introductory proof-based classes. Others may offer a proof-based approach to linear algebra, and more rigorous curricula may even include things like functional analysis. While all of these subjects form the bedrock of higher mathematics, it is my opinion that none of them is well-suited to teaching proofs per se.
In a previous post, I explained why proofs are so difficult to understand. In this post, I'd like to explain how to think about them productively.
A proof takes given axioms, theorems, and other mathematical properties and combines them in a way that demonstrates the truth of a new mathematical property. The first step in conquering proofs is to acquire a new mindset about solving problems, one that is better suited to writing proofs rather than solving equations. The next step is to understand exactly which logical moves are permitted by the given knowledge, and just as importantly those moves that are not. The final step is to acquire a toolbox of proof techniques that enable you to more easily solve an ever-growing suite of problems.