I disagree with all of these recommendations about studying analysis.
tl;dr: Study a less careful subject where you can more easily develop a useful intuition. Abstract algebra and linear algebra both fit in that category.
(What follows is a caricature.) I tell people that analysis was born in the 19th century when Weierstrauss killed intutition as a tool for understanding calculus. Analysis might teach you what is "really true" but unless you are going to be some kind of professional I would HIGHLY recommend developing an intuition and understanding of things based on what should be true. It's a terrible thing to learn that what "should be" true is not, and for a long time that doesn't help you make progress.
This is because analysis is built upon the fantasy axiom of "real" numbers (of which, according to the measure theory of real analysis, 100% are impossible to even have names, even in principle, let alone compute with), instead of the solid foundations of algebra.
tl;dr: Study a less careful subject where you can more easily develop a useful intuition. Abstract algebra and linear algebra both fit in that category.
(What follows is a caricature.) I tell people that analysis was born in the 19th century when Weierstrauss killed intutition as a tool for understanding calculus. Analysis might teach you what is "really true" but unless you are going to be some kind of professional I would HIGHLY recommend developing an intuition and understanding of things based on what should be true. It's a terrible thing to learn that what "should be" true is not, and for a long time that doesn't help you make progress.