I agree with Adrián that Rudin, and analysis generally, is not a good first exposure to proofs. There are at least three subjects I can think of off the top of my head that are much more accessible for such a thing:
- Elementary number theory
- Elementary graph theory
- Elementary combinatorics
In these subjects the objects one is proving facts about are much easier to grasp intuitively. I don't know good references at the introductory level off the top of my head, but you might try telling your friend to browse the Art of Problem Solving books.
I have to warn you that your estimate on the amount of time to finish Rudin (if done correctly) may be off.
Here's why. Up to now, you've taken the standard advanced course in high school mathematics and done quite well. This is a feat to be proud of, and unfortunately, you've done so well that you are a year ahead of the game. I say unfortunately, because the next natural step would be to take a proof based math class and learn the fundamental skill of writing clear, coherent mathematical proofs. It doesn't matter the subject through which this is done, but this is the step that should happen next.
The problem is this next step is difficult (if not detrimental) to take alone. You need someone to read your proofs, to make sure your arguments make sense and are understandable to another person, and to check that your sentences end in (goddamn) periods.
You can't do the exercises in Rudin (and for that matter learn basic analysis)
without having the skills of proof writing. And for that reason, I advise you to try to find someone to help you acquire this skill. Here are three ideas.
(1) Where are you from? There may be math classes at a local university you can take and get credit for. This will have the added benefit that you will meet other people who like math. Talking about Math is a lot of fun. And while, many mathematicians learn a great deal through self study, it's typically in the context of a mathematically inclined environment. It might be surprising to learn how much of the stuff you think you know is wrong when there is someone there you try to explain it to.
(2) If that fails, try to find a correspondence course. This way you at least get feedback and keep the postal service afloat.
(3) Find a teacher at your school. Many (maybe all) were probably math majors at one point, and could read over your proofs and give feedback.
However, if none of these options are available, I would advise you to stick to the more computationally minded brand of mathematics that you have seen in calculus and differential equations. There are great treatments of linear algebra in this vein. Try Gilbert Strang's 'linear algebra and applications' which has an associated lecture series on MIT open course ware. Another option is to try to learn some programming. Java's great. And tackling a programming problem will stimulate you in a way you might have once thought was reserved only for mathematics.
If all else fails. Fly a kite, learn to surf, and prefect a secret BBQ sauce recipe. It's your last year of high school! Live It Up.
Best Answer
Rudin can be a bit terse. Here is an alternative: Introduction to Real Analysis by Bartle . It's a good book. It dosen't cover Dedekind cuts though, if that's what you're struggling with (you said that you just started reading it, and the first part of the book is on Dedekind cuts). Anyways, read what you like. Some people like algebra more than analysis. Some are the other way around. Don't think that you have to read from specific famous books. People have learning styles, and this is one reason why we have many different books on the same subject.