What can one learn from real analysis that is not taught in high school?


There's no universal standard for what “real analysis” means as an undergraduate course, nor for what you learn in high school. Typically, the main difference between a high-school calculus class and a college-level course in real analysis is the presence of proofs in the latter, alongside formal definitions. The “new skill” you should expect to acquire isn't the ability to calculate more derivatives or more integrals. It's becoming familiar with the idea of proof, an appreciation for why it's needed, and practice in coming up with proofs and writing them down. This is a significant challenge for many students, but it doesn't have to be, and it's a very worthwhile endeavor.


Answer by Alon Amit, PhD mathematics, on Quora.