I'm in my 5th semester undergrad for bachelor of engineering in mechanical engineering, but I realized that I like mathematics much more than I enjoy my ME or any other applied courses that I have to take. I'm thinking about applying for graduate study in mathematics at top US schools (Berkeley, MIT, NYU, etc), but I'm afraid that I'm ill prepared and that my efforts will be for nothing, at least for those better schools. On paper, I have no advanced math courses completed. My coursework includes Calculus I and II, and multivariable calculus, differential equations, and probability and statistics. It gets worse because we were using baby texts such as Stewart's Calculus Concepts and Contexts for Calc I, II and multivariable, and not-so-bad Differential Equations by Snider and Saff. Additionally, there was no emphasis on theory whatsoever, just lots of blind computation and plug-and-chug. Nevertheless, I always read the textbooks on my own and considered the proofs to be essential and tried to follow the arguments given to the best of my abilities, even though I didn't quite understand why a proof by contradiction works, or any sort of indirect proof actually. Things turned when I tried to go for a minor in EE and enrolled in a class called Math for EE, which basically was divided into two parts: First half was a survey of Discreet Mathematics (emphasis on propositional calc, predicate calc, valid rules of inference, valid proof methods, common fallacies, naive set theory, graph theory) from the book Discrete Mathematics by K. Rosen, and a second part was intro to complex analysis from a text by Brown and Churchill. I had to drop the course 2/3 in because I was overloaded with coursework, so I can't really add this to any graduate application but at least the knowledge stuck. Right now I'm working myself through (self-study) Calculus vol. I by Apostol and I'm doing every proof excercise and I find that I have a talent for this, but these are still just baby steps. I try to read slowly, reflect on the proofs, and spend time on the excercises, but time is one thing I'm running out of because soon I have to decide if I want to enter the workforce or continue onto graduate level. Any math major will probably blow me out of the water because I've looked at top undergrad math courses and I see courses such as abstract algebra, topology, differential geometry, and many many others that I just won't have time to get myself properly acquainted with before I graduate. Basically, I'm trying to establish whether graduate mathematics is out of my reach and a hopeless goal? I'm asking this on an internet forum because my parents just aren't able to help me this and I have never gotten a sound advice from counselors, but this board seems populated with some well qualified people.