Keepin' it real: Real analysis for the mathematically lacking
So, you want to know what real analysis is all about? First of all, a warning: This is one of the two "hard bits" in the switchover to university level mathematics1. This is the first introduction to truly rigorous proof, and all the headaches that come with it. I'm going to try and keep the level of tediously anal argument to a minimum, but it really is necessary, at least until a certain point (generally about midway through your second year of study, unfortunately). With that out of the way, let us proceed to the main question.
What is real analysis and why do I care?
A very good question, Mr/Madam reader sir/ma'am. Real analysis is the study of functions on the real line. It's about what makes the real numbers special, as opposed to the rationals or the complexes (complices? I'm not sure). As for why you should care, this is where the question gets interesting. Really, you shouldn't. It's possible to do most, if not all, of the mathematics you'll need for any kind of physical application, without knowing the innards of a real analysis course.
However, analysis in general, and real analysis in particular, are very important in another aspect. They provide a rigorous logical foundation for everything that comes ahead. While a lot of it falls under the category of "proving the bloody obvious", this is still important. A first course in analysis (usually a real analysis course) will take the concept of proof to a whole new level. Without a background in real analysis, areas such as complex analysis and topology are incomprehensibly difficult. While the basic concepts are understandable, the mindset required is completely alien.
So where do we start?
We start by asking what makes the real numbers special. To work this out, we need to look at their closest cousins, the rationals. The main distinction between the two is this:
Consider the sequence of numbers 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... This sequence converges (in both the intuitive and rigorous mathematical senses). However, although all the numbers in the sequence are rational, the limit (π for the slow on the uptake), is not rational. Try doing something similar with the reals. Try finding a sequence of real numbers which converges to something that isn't real. You can't. Trust me. This property of a set is known mathematically as completeness.
So wow. The reals are complete. You haven't actually done anything, have you?
Well, no. Not as such. But, using some hard work and a bit of magic not reproduced here for reasons of brevity, we can show that the Completeness Axiom, as our favourite new fact is known, is equivalent to two other statements. The first of these, the supremum axiom, states that any set of real numbers with an upper bound has a least upper bound. Again, note that this is not true in the rationals (there is no least upper bound on the numbers whose square is less than 2, for example). The second, and most widely used reformulation is the terrifyingly named Bolzano-Weierstrass property. This is the statement that any bounded sequence of reals has a convergent subsequence (with a limit in the reals). Harder to understand and see intuitively, yes, but a lot easier to apply in proofs
So, armed with our new weapons, we set out to prove our first piece of obviousness. This is the famed Intermediate Value Theorem. I'm going to set out the statement of this in full, just so you can see how utterly moronic it really is.
Theorem: (Intermediate Value Theorem) Given a continuous function f: [a, b] → R, with f(a) < 0 and f(b) > 0, there exists c ∈ [a, b] with f(c) = 0
Or, in plain English, if you start below a line, don't make any jumps and end above the line, at some point you were on the line. Of course, the mathematical definition of continuity means that this requires proving. To do so, we apply the supremum axiom. We look at the set of numbers between a and b such that f(x) < 0. This certainly has an upper bound, say b, so it must have a least upper bound, which we'll call c. We can now fiddle about with this number in some silly way and show that f(c) = 0, like we said.
Hooray.
Well done. You've told me something I already knew.
Perhaps. You're probably right. But lots of pure mathematicians would argue that unless you know how to prove something, you can't truly know it. There must be at least some niggling doubt that it might not be true. Now you can lay that doubt to rest. And you can go on to prove lots of other things.
Prove other things? Like what?
Ooh, lots and lots. I haven't even defined a derivative yet. You know all that calculus you were told at school? You get to see where it comes from. Things like the chain rule which everyone takes for granted, you can actually derive and prove. You can tell exactly how silly things need to get before everything you know falls apart and what to do when it does. And of course, you lay the groundwork for later, much more interesting work such as topology and Fourier analysis.
First, however, there's a pretty long slog through Rolle's Theorem, the Mean Value Theorem, Fubini's Theorem, Taylor's Theorem and lots more in order to get a base from which to work. A lot of it is dry and dull, but it provides a basis for a kind of thought that runs throughout mathematics. And, more importantly, it helps reassure some of us that the universe is still working correctly.
What I've given here is a pretty vague and woolly account of a very rigorous discipline. To any analysts reading, you have my sincerest apologies. To any would-be analysts who find what I've said intriguing and want to know more, the traditional introductory text is Burkill's A First Course In Mathematical Analysis. Also possibly of use is A Companion To Analysis, a text available online at http://www.dpmms.cam.ac.uk/~twk/, the website of Professor Tom Körner, the man who instils Cambridge undergraduates with a healthy respect for the subject. Even if you don't understand it, read it for the wonderful prose.
1The other one is group theory, which introduces the idea of totally abstract structures.
Part of Maths for the Masses