In addition to everything else you do, I recommend the following book:
"Counterexamples in Analysis" by Gelbaum and Olmsted.
You will find that many of your intuitions you picked up in calculus are violated in analysis. For instance, in calculus, many examples are both continuous and differentiable everywhere. But is every continuous function also differentiable? Nope! See the Weierstrass function [0].
The book is full of such counterexamples that will help you understand analysis at a deeper level (and avoid many pitfalls)