I heard from a renowned Mathematician that self study of proof based Mathematics is extremely difficult as there is not only right and wrong but also degree's of correctness. So without a teacher actually guiding you through the subject it's very difficult to analyse our mistakes.
If this is true, what should one do to overcome this problem?
By proof based I mean Analysis, Topology etc..
All mathematics is "proof based" because all mathematics are supported by a logical structure. We can focus on the structures to varying degrees.
When I think of "proof-based mathematics" the first thing that comes to mind is a "classical" high school geometry class. Except for top private and public schools, in the US most students no longer take such a course. Most people think that "real world" problems help students connect to the material. This is why modern primary math texts are cluttered with more photos than a pop star's Twitter feed.
Students who are being groomed for more serious education in math or science are far more likely to have access to a course like classical geometry-- with the right teacher calculus can serve the same function.
Classical high school geometry was designed in the hopes of being the easiest possible introduction to writing proofs. With less content and very few calculations to perform students were meant to focus on the logic. For students who view math mainly as a means of calculating answers, this can be a difficult transition.
Like many people who love math, I adored this course. I also found it very easy since I struggled with accuracy in calculations. Despite its simplicity, it was the course that prepared me for topology and analysis in college. It was more helpful than even calculus (which, in theory, can be proofs based... but many iterations make it easy for students to ignore this aspect.)
I think it could be very difficult to encounter your first "proofs based course" in the guise of analysis I even with an instructor. I suspect this is what has been happening for many students for years now.
And some of them make it, so it is not insurmountable.
I still think that it is easier to learn about the concept of proof with material that less abstract than the courses you mention.
As for "degrees of correctness" -- this is a bit confusing, There are often many ways to prove a theorem or solve a problem. More than one way can be correct, but often one way has fewer steps or is easier to understand.
But, each is either correct or incorrect.
But maybe you could say more about this?