I use "Proof: An Introduction to Higher Mathematics," by Esty & Esty (my father and me). We self-publish, in order to keep it relatively cheap--I think bookstores sell it for about $45, depending on the markup. The chapters are split into two categories, theory and practice:

(Theory)
1: Intro to proofs (sets, logic)
2: Sentences with variables (generalizations, existence, negations)
3: Proofs (inequalities, absolute values, contradiction,
contrapositive, induction)

(Practice)
4: Sets (set theory, bounds)
5: Functions (one-to-one, onto, functions on sets, cardinality)
6: Number Theory
7: Group Theory
8: Topology
9: Calculus

At the liberal arts college where I teach, we generally get through the first five chapters (in a one-semester course). One could skip around more than I do, however.

The main thing our book does differently than others is emphasize a lot of common grammatical mistakes students make when first learning proofs. We found a lot of proof books already assumed that students understood a lot about the language we use when we write proofs, and only taught specific techniques like induction. We spend more time on the language at first, including conventions as well as logic. Probably for strong students who already possess good mathematical intuition it would be unnecessary, but we've found it works better for our students. The book has been used at Montana State, Marshall, Case Western, Boise State, Texas State San Marcos, etc.

If you think you're interested, there's more info here:
http://estymath.com/Proof.html.

It seems some faculty want proofs in combinatorics and equivalence classes. Our thoughts were that we wanted to prepare students for classes with many definitions of terms and proofs using them, such as Advanced Calculus, Real Analysis, Linear Algebra, or Abstract Algebra. Combinatorics has a method of proof all its own that is not seen much in those classes, so we omitted it, and we do only a bit of equivalence classes because they are short and easy given what we cover. So, if you need a lot of combinatorics, our book is not the right one for you. If you are at a high-powered school with very strong students our book is not the right one for you. However, if your students make the same sort of logical and grammatical mistakes commonly seen in "Introduction to Real Analysis," this text may be right for you.

I teach at Stonehill College, where we have a proofs course called "The Language of Mathematics," which is taken by math majors after Calc II, concurrently with Calc III. We introduced the course a few years back because we found that students weren't really prepared for the rigors of analysis or algebra, and that a lot of time was being spent in all upper division courses teaching the same stuff. Things are definitely better since we've added the course.

separatecourse is needed is quite a feat in absurdity! $\endgroup$and they're different skills.IMHO, combinatorics is an excellent subject for learning to write rigorous proofs, precisely because the definitions are easy to understand, and you don't have to spend a lot of time proving theorems which "look obvious". $\endgroup$8more comments