I have long thought of real analysis as the consequences of mapping numbers to real numbers, thereby creating infinite sequences and series. From there, I can get to fairly elementary applications like differentiation, continuity, Riemann integration, up to improper integration.
But then I learn that the scope of real analysis also includes compactness, measure theory, Borel sets, Lebesgue integrals and things of that sort.
How does "infinite sequences and series of real numbers" connect to the kinds of things discussed in paragraph 2? Put another way, how do you use sets to "cover" terms of infinite sequences, and why would you do this anyway?