I wasn't a math student, but I would probably look at the OpenCourseWare from MIT if I were trying to learn this stuff.

Analysis is 18.100 at MIT -- the variants are called 18.100A, 18.100B, 18.100C. There are further classes in the same vein, as well, such as 18.101.

https://ocw.mit.edu/courses/mathematics/18-100a-introduction...

https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-f...

https://ocw.mit.edu/courses/mathematics/18-100c-real-analysi...