As I found Mathematics the most supreme beauty, a true Paradise, I started and committed to Mathematics. [...] As I was keen to have a big impact on Mathematics, [...]
This is quite off-putting. What you say here is essentially that you want to be a great mathematician. Since there is no a priori reason to assume that you are better than many other students the professor you address has, such a statement is a little presumptuous.
I had a consecutive study plan and I have studied ... [name of books].
This is good. You give a more or less objective description of your abilities and show that you actually have done serious work.
What's missing is a description of your actual interests. If you say you are interested in mathematics or real analysis, you essentially say that you have not found something that really intrigues you. As research is largely internally motivated, this is not a good sign. Note that I do not say that you have no special interests, but that what you write in your question and what you state as your approach to a professor says so.
So a letter which should attract the attention of a professional mathematician could be the following:
I am (description + why you have no direct contact person).
I have read the following books.
I am fascinated by (special topic), because (honest reason). In particular, I would like to understand (something you are really interested in).
Can you recommend me further directions for my studies, e.g. textbooks dealing with (whatever)?
If you get a reaction like "read XYZ", and after half a year you pose a question which shows that you have worked through this book, you will probably be taken seriously.
