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.

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.