Now that I've developed enough topology to prove that the real numbers are not countable, I'm ready to move on to the next topic, which is sequences and series. Before I get into the details, I want to talk about some general issues around the subject.
First off, the book specifically develops topology before sequences because it requires topology concepts in order to provide the structure for discussing sequences. On the other hand, I ended up using sequences frequently in my discussion of topology. I didn't rely on a lot of theory of sequences, but I did use an implicit definition of sequences. If you read some of my proofs revolving around sequences of sets and were wondering how exactly sequences were defined, I will clarify that now. At the same time, there were certain proofs where I desperately wanted to use more theory of sequences, and I avoided doing so because I hadn't developed the theory yet. I don't think greater use of sequences would have made my posts about topology more rigorous, but it may have made it easier to understand.
I've previously mentioned the idea that studying any particular subject in math leads inevitably to studying every subject in math. The way that sequences and topology are tangled up with each other leads to the conclusion that not only do you have to study every subject, but you have to study every subject at the same time. They all build each other. As another example, sequences and series are usually introduced as topics in Calculus 2, when you can use concepts based on integration to prove the basic theorems. On the other hand, the proof that integration works is dependent on Riemann sums, which are infinite series. This sort of thing could turn into circular reasoning, but it can also turn into both topics providing scaffolding for the other, where the results are rigorous but dependent on developing both subjects simulateously.
Speaking of Calculus 2, when studying sequences and series in Calculus 2, I felt like the material both didn't do enough and was looking to do too much. The important question is whether or not a particular sequence or series converges, not what that sequence or series converges to. This was frustrating to me, because it feels like we're only solving half the problem. At the same time, the tools for determining convergence feel weak, making applying them difficult. It's like we're choosing to only solve half the problem, and then handicapping ourselves to make the problem hard.
The approach we are now taking to sequences and series doesn't require calculus, which is good. But it doesn't really prove any more than calculus did, which is bad. We're still just asking whether a sequence converges, and we still end up with a similar set of tools. At least the results are consistent with Calculus 2, so there's at least that.
As I've been studying analysis, I've become convinced that sequences are the basis for everything that follows. Even when the presentation feels incomplete or weak, it's important to understand sequences in order to support analysis as a whole. So this will be my next major topic on this blog.