Aiming for a Pure Mathematics Degree.
So I have currently completed 1.5 years at a junior college and have completed all of my lower division mathematics courses (i.e Single Variable Calculus, Multi-Variable Calculus, Vector Calculus, Linear Algebra, and Differential Equations).
As of right now I am still waiting on a response from a university to see if I am accepted for their Spring semester. If I do get accepted, I will have 3 months off from school. If I am not accepted, then I will complete my spring semester at my junior college, but I will not be taking any math since the courses I've completed are all they offer.
So my question is, what do I do for self studying? I have bought a couple math books but I am unsure on which one would be the best place to start. Any help, guidance, or recommendations for other books would be greatly appreciated.
The books I have are:
"Introductory Real Analysis" -A.N Kolmogorov
"Advanced Calculus" - Patrick Fitzpatrick
"Introduction to Topology" - Bert Mendelson
"Linear Algebra" - Georgi E. Shilov
"Ordinary Differential Equations"
Also, how can I improve my "proof writing" ability? It is something that I had a lot of trouble with in Linear Algebra.
Start with a book aimed to help you improve your ability to prove things. Proofs can be a significant road block for many people in learning new material because they lack the ability to verify statements/theorems for themselves, and then depend on others to walk them through it. It is also a good idea while reading to try and prove theorems before you read their proofs in the textbook. Doing this will give you an idea of what previous statements are important for the theorem, and also give you a little bit of exercise in using these new tools. But if you are lacking in the proof department, this will require far more time than necessary.
Once you are strong in your proofs I would recommend picking up any book that interests you (which you also have sufficient background to understand), and working through it. You can try working through a discrete math textbook (if you haven't already taken it), or "Linear Algebra Done Right" which seems to be an appropriate book for those just beginning.
Good Luck.