I had been posting about real analysis. I worked through the definition of the real numbers, then moved on to topology and sets, and finally had started working on sequences and series. Then I stopped. Truthfully, when I started covering this material I hadn't intended to cover things in quite so much detail, but going over all the details did come pretty naturally to me. But when I started, I had been intending to keep pace with my own studying. I ended up falling behind, and then I ran out of time for any blogging at all.
In the meantime I kept studying, and now I have worked through two semesters worth of analysis. While I'd like to return to blogging in detail, I've decided to take advantage of the fact that I've stopped to take a step back and look at the larger picture, now that I have enough understanding to see the larger picture.
The fundamental idea behind real analysis is the limit. Limits are a powerful tool for understanding lots of math concepts, and real analysis is about developing the use of limits and then applying them to various problems.
Loosely speaking, the idea behind limits is that two things are near each other. The things in question could be numbers, or points in space, or sets, or functions. There's a precise mathematical definition for limits, which involves Greek letters (and causes some people to run in terror), but today I just want to talk about the general concept.
In many mathematical contexts, the standard is exact equality. High school algebra is all about showing that the left hand side of an equation is exactly equal to the right hand side. With limits, we say that two things are not exactly the same, but that's okay as long as they are near each other. This can feel like it's a step back from true equality, and it can also feel unfocused.
But there's a tradeoff. Equality can only say that this thing is exactly the same as this other thing. Limits can let you say that everything near this thing is close to everything near this other thing. The ability to speak about lots of things at the same time gives limits more power than strict equality has.
You may be wondering why, if analysis is all about limits, did I spend months blogging about sets and sequences. I did not know the answer at the time, but now I do. Just like limits are a tool used by analysis to talk about other stuff, we need tools to talk about limits. The first tool is sets and topology. One of the fundamental concepts of topology is distance, and so gives us the ability to talk about whether two things are near each other. The theory about sets we developed, for example the properties of compact sets, gives us tools to talk about limits.
Similarly, sequences give us different tools to talk about limits. The important thing here is that although sets and sequences give us different tools, they come to the same conclusions. Anything that can be demonstrated about limits using sets can also be demonstrated using sequences, and which one to use is just a question of convenience. This equivalence can also be used for sets and sequences to say things about each other, so using both tools allows us to get a deeper understanding of each tool individually.
I plan to have another post soon in which I will talk about what limits are useful for, again at a big picture level. I may also post about the big picture with sequences and series. My introduction to the concepts of sequences and series was in Calculus 2, and the idea has always felt a little half-baked. Now that I'm looking at them from the other side of analysis, I have a much better understanding of why we study them the way we do.