I strongly urge you to read an answer that I wrote up for another question: (link).
The landscape of higher education is changing rapidly right now, and the path to becoming a tenured professor is not very similar anymore to what it was even just 15 years ago. I believe (others will disagree) that this process will change even more dramatically in the next 15 years.
If math is what you love, you should definitely continue studying it and pursue it to see what options are available to you. But don't disregard the need to assess your aptitudes and honestly appraise your skill next to the skill of other PhD quality mathematicians vying for the same faculty jobs.
Along these lines, I personally think it's better to go for an applied course of study. Devote time to learning excellent programming skills in multiple languages, as well as non-trivial software design skills. Knowing how to tinker in Matlab, Maple, and Mathematica is not worth anything. Similarly, learn advanced statistics. Study what people do with large data sets (mostly computational Bayesian methods these days). Learn about scientific computing and implementational details.
Additionally, choose a topical domain for which you believe the job outlook will be good. This could be computational finance, computational biology, applied machine learning, or a host of others about which I am less familiar.
Ask your current professors for advice on this. But be careful. People who were lucky enough to make it to the position of professor often suffer from narrative fallacy and selection bias. That is, rather than acknowledging that they are not much more skillful than peers who were not able to win faculty jobs, professors tend to attribute their fortunate position to various narrative stories about what they specifically did to work hard and achieve things. But what worked for one person in one situation is too idiosyncratic for you to care about; it doesn't describe what works for general situations, nor for the future situation that will be relevant to you.
Do lots of research before committing yourself to one direction or another. And consider many other important life factors, not just how much you like math, such as:
- Do you want to have a family? Children?
- How important is salary for the lifestyle you want to lead?
- How important is geography? Young professors rarely get to pick where they live.
- How employable will your skill set be if you don't get tenure and/or cannot find an academic job?
These things matter a great deal in your decision in college to orient yourself towards a future career. Most people will give advice to you in far-thinking mode, but this isn't a good thing. You should realize that the economic landscape of the world in 5-15 years will determine what jobs you have the option of doing. That's just an attribute of reality. And the more time you spend reflecting on that and planning for what reality will be like, the better suited you'll be to try to make your own goals happen. And, more importantly, the better capable you'll be of re-orienting your goals to match with what is possible.
Best Answer
The only prerequisite is the proverbial "mathematical maturity". To be a bit more specific, you should be comfortable with the basics of naive set theory. Just to cite a few examples:
That's not meant to be a comprehensive list.
There's a famous book that covers the construction of $\mathbb{R}$ from $\mathbb{N}$: Landau's Foundations of Analysis. It's written in a severe Definition-Theorem-Proof style that people used to admire. Personally, I don't recommend it for beginners.
For a more user-friendly source, I'd recommend Enderton's Elements of Set Theory. This covers the preliminary material you need, before constructing the natural numbers in Chapter 4 and the integers, rationals, and reals in Chapter 5. (Probably you can find a free pdf of the whole book online, but with a quick look I've found only Chapter 5.)
I should also mention Dedekind's original paper, "Stetigkeit und irrationale Zahlen" ("“Continuity and Irrational Numbers"), giving his construction of $\mathbb{R}$ from $\mathbb{Q}$. Dedekind is a remarkably clear writer. Of course, his notation is outdated in places, though when I read it (many years ago), I don't recall that causing any hiccups. However, looking for an online copy (available at Project Gutenberg), I ran across this paper: An Examination of Richard Dedekind's "Continuity and Irrational Numbers", Rose-Hulman Undergraduate Mathematics Journal, by Chase Crosby.
Yet another source from a very good writer: the epilog to Spivak's Calculus. He gives three constructions of the real numbers, namely Dedekind cuts, Cantor's fundamental sequences, and infinite decimals. The second two are presented as exercises with detailed hints. He also shows the essential uniqueness of the real numbers.
Three more remarks: (1) Category theory sheds new light on some of this, specifically what "essentially the same" means. (2) The construction of $\mathbb{N}$ as von Neumann ordinals can seem rather artificial. What really matters is not what natural numbers are, but the axioms they obey. (3) This is true also for the later stages of the construction. Specifically, there are two famous constructions of $\mathbb{R}$ from $\mathbb{Q}$: Dedekind cuts, and Cantor's fundamental sequences. They look pretty different, but you end up with isomorphic structures.