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  • What is Mathematics, An Elementary Approach to Ideas and Methods - Courant Robbins Stewart
  • How to Solve It, A New Aspect of Mathematical Solving - Polya
  • Introductory Mathematics, Algebra and Analysis - Smith
  • A Transition to Advanced Mathematics - Smith Eggen St.Andre
  • An Introduction to Mathematical Reasoning - Eccles
  • A Book of Proof - Hammack
  • How to Prove It, A Structured Approach - Velleman

i.e. by "Transitional" I mean a book that can take the reader from college Mathematics to the more rigorous and abstract Mathematics taught to Math Majors

i.e. by "college Mathematics" I mean all the scientific majors that include Mathematics in the curriculum, Mathematics Major excluded. Only Math students take on Honors Math programs that are quite different from other Math courses (generally more problem-solving-based) for rigour and proofs. Although some books are suggested at Cambridge and Oxford, the question is grounded on the US education system - more than the European or any other system in the world.

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    $\begingroup$ How to prove it. $\endgroup$ – Git Gud Jun 4 '14 at 21:50
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    $\begingroup$ @Ephraim, actually "college math" is often the name for a course or textbook aimed at non-majors. $\endgroup$ – StumpyLeg Jun 4 '14 at 22:14
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    $\begingroup$ You really need to be more specific about where you are coming from - "college mathematics" can mean a lot of things. $\endgroup$ – Thomas Andrews Jun 4 '14 at 22:14
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    $\begingroup$ I've done problems from Spivak's Calculus, and many were even tougher than the book I used for real analysis. I would say that some book that primes your proof techniques maybe necessary rather than jumping into it. $\endgroup$ – user60887 Jun 4 '14 at 22:24
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    $\begingroup$ Feel free to add to the presented list of books if you deem it absolutely necessary. Like "Mathematical Proofs, A Transition to Advanced Mathematics - Chartrand Polimeni Zhang" which I haven't mentioned. @vonbrand Thanks. I do have access to all the books I've enumerated $\endgroup$ – ex0plan Jun 5 '14 at 10:39

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