Soft question regarding topics In real analysis

Question

I'm self studying "Understanding Analysis by Stephen Abbot". I'm starting to enjoy the subject, but found it slightly challenging from the first chapter which I've now completed. Do the topics on sequences, series, continuity, integrability that follow in an analysis subject become better to manage and somewhat easier to think about intuitively after studying the place of the Real number system.

Answer 1

Like anything else, intuition comes through experience and practice; you will not immediately get it. Real analysis is not a trivial subject either; which is ironic because, at least the way it is often taught at the undergraduate level, it's essentially making calculus more rigorous, which is a subject we introduce to high schoolers.

It's an example of the fact that while the ideas that come from it may be simpler and easier to grasp, the underlying framework is very nontrivial and can indeed be difficult to understand. As a matter of perspective, at least at my university, it is generally regarded as the hardest class for math majors.

So don't be surprised if you're not getting it - it's not something a ton of people can get, so if you're understanding stuff so far, even if you don't yet have the underlying intuition, that in itself is something to be proud of. The intuition won't come immediately, but it does come about if you practice and read thoroughly through the material. Personally, I also found self-explaining proofs to be really helpful; granted, this was in the university setting and not self-study, but even just rewriting down proofs from the text or lectures, in my own words, explaining in my own way, went far in helping me to understand the material and develop an intuitive understanding of what went on in the course.