As a general reflection on undergraduate mathematics imho there is way too much emphasis on real analysis. Yes, knowing how to be rigorous is important, being aware of pathological counterexample is importanting, and real analysis is used all over the place. But there is so much more to learn in mathematics than real analysis and the focus on minor technical issues here is often a distraction to developing a broad & deep mathematical background.
For most mathematicians (and scientists using serious math) real analysis is a only a small part of the toolkit. Understanding well the different kinds of limits can ofc be crucial in functional analysis, stochastic processes and various parts of physics. But there are so many topics that are important to know and learn here!
The reason it is so prominent in the undergraduate curriculum seems to be more tied to institutional inertia, its prominence on centralized exams, relation with calculus, etc
As a general reflection on undergraduate mathematics imho there is way too much emphasis on real analysis. Yes, knowing how to be rigorous is important, being aware of pathological counterexample is importanting, and real analysis is used all over the place. But there is so much more to learn in mathematics than real analysis and the focus on minor technical issues here is often a distraction to developing a broad & deep mathematical background.
For most mathematicians (and scientists using serious math) real analysis is a only a small part of the toolkit. Understanding well the different kinds of limits can ofc be crucial in functional analysis, stochastic processes and various parts of physics. But there are so many topics that are important to know and learn here!
The reason it is so prominent in the undergraduate curriculum seems to be more tied to institutional inertia, its prominence on centralized exams, relation with calculus, etc