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Can anyone suggest some good books to help an high school student to "bridge the gap" to university math? I've heard of http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316 and http://www.amazon.co.uk/Bridging-University-Mathematics-Edward-Hurst/dp/1848002890 Do you have any other suggestion?

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I wouldn't read any book like this. you don't need a guide for studying university mathematics, you just do it. You are just wasting the time in which you could do actual mathematics (no offence, its merely my opinion). –  Alexander Grothendieck Feb 16 at 14:02

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Permit me to suggest a wonderful book that will be immensely helpful to you:

How to Prove It: A Structured Approach, by Daniel Velleman.

Here is an editorial "blurb":

"Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs."

If you follow the link above, you'll be able to "preview" the book, including its table of contents.

Aside: One feature I personally love about this book is that it is both "about math" and "is math": You learn about more mature approaches to math and proofs by doing mathematics, proofs and all.

I'd suggest previewing Mason, Burton, and Stacey's Thinking Mathematically, as well, since many students find it to be a great aid in "transitioning" to a more mature appreciation of and approach to mathematics.

Another classic which many highly recommend is Polya's How to Solve It. Again, you can "preview" this book if you follow the given link.

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