What is Real Analysis?

If you've taken Calculus, you will be familiar with terms like limit and continuity. In Calculus itself, you spend a lot of time with differentiating and integrating, and you use the ideas of limits and continuity to be able to do those things. However, Calculus is a word that you see used by scientists and engineers, where as mathematicians refer to Calculus as being one part of a larger subject called Analysis. Real Analysis simply refers to focusing the ideas of Analysis on functions of real numbers.

Now, if you were to ask a Calculus student what it means for a sequence, let's say, to have a limit, what do you think his or her answer will be? He or she might say, "When the terms of a sequence get really really close to a number, that number is the limit." For Calculus, this might be an acceptable description, but "really really close" is not very rigorous and when you get into Analysis you need to be able to describe how close you need to get to that limit, and how to get that close to the limit. Consider the two following sequences:

an = 1/n = 1, ½, ⅓, ¼...
bn = 1, 0, 1, 0, 0, 1, 0, 0, 0, 1...

As you can see, the terms in an are getting closer to 0 as n increases, and the terms in bn are 0 much more often when n increases. With the definition of a sequence's limit being when the terms get "really really close to a number," you could say that the limit of both sequences is 0. It is true that the limit of the first sequence above is 0, but in actuality the second sequence does not have a limit. Let's take a look at a more rigorous definition of what a limit is:

A sequence an has limit L if for all ε>0, there is an integer n0 such that for all n>n0, |an - L|<ε.

Now, if you're seeing this definition for the first time, it probably doesn't make much sense to you. For one thing, you probably have never used the Greek letter Epsilon (ε) before, and even then, what is n0? What this limit tells us is that "really really close" means we can get as close to that limit as we want. The ε term is an arbitrary positive number, meaning that we can pick whatever value we want for it (usually we like to pick small values), and once we pick our ε, we need to show that there is some point in the sequence (n0) where the terms from then on are going to be at most that far away from the limit.

For example, in an as described above, if we were to let ε=½, one possible value for n0 would be 2. So, for all n>2, |1/n - 0|<½. Similarly, you could let ε be as small as you want as long as it's still positive, and you can find an appropriate n0. If we were to look at bn as described above, we can show that there is no limit. Let's assume to the contrary that the limit of the sequence is 0, and once again set ε=½. While for large n, an is going to be 0 most of the time, there are still going to be terms that equal 1. So, there is no n0 that will satisfy |bn - 0|<½, so we cannot get as close as we want to the limit.

So, this definition of limit answers the two questions that I proposed before. How close do the terms in the sequence get to the limit, and how do you get that close? The answer to the first is that the terms get arbitrarily close, and you can show that the terms get that close by showing that there's a point where the terms stay within that close range.

Now that we have discussed the idea of a limit, we can use similar ideas to more rigorously define what it means for a function to be continuous, what it means for a function to have a derivative, and what the Riemann Integral actually is.