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As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an advanced calculus course, not a real analysis course). In the fall I will take the first of a two part course in advanced calculus. I believe we will use Fitzpatrick's book, Advanced Calculus.

So, in order to prepare for the rigor and proof writing that will be necessary in this course, is there a good book or pdf that will provide solutions to problems involving proofs? I've found plenty of books with tons of problems but finding solutions to check myself (or see if there is a more clever approach) has been difficult. I hope this isn't too broad of a question, maybe some of you coming from smaller universities will understand my dilemma.

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    $\begingroup$ I think there are two fundamental steps to writing good proofs: 1) Knowing and understanding the logic required to complete the proof; 2) Reading a lot of proofs. $\endgroup$ – anakhro Jun 11 '15 at 17:51
  • $\begingroup$ That's great, care to share these steps? $\endgroup$ – user247518 Jun 11 '15 at 17:52
  • $\begingroup$ Sorry, didn't see the full comment until now. $\endgroup$ – user247518 Jun 11 '15 at 17:53
  • $\begingroup$ Jim Franklin, a professor at my university, wrote a book entitled "Introduction to Proofs in Mathematics", which can be found on Amazon. He was my tutor for linear algebra this semester, and he taught me a lot. $\endgroup$ – eloiprime Jun 11 '15 at 17:53
  • $\begingroup$ Do you know of any books or pdf's that would help me practice those steps? $\endgroup$ – user247518 Jun 11 '15 at 17:54
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Analysis With An Introduction to Proof, 5th Edition by Steven R. Lay. I thought this was a great book to learn how to write mathematical proofs. Shows you completely how to write proofs and the approach to take in how to start a proof.

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I like the book Mathematical Writing by Franco Vivaldi, based on a course on writing offered to second year undergraduates at Queen Mary, University of London. It used to be offered online for free but seems to have been snapped up by Springer. Perhaps your library has a copy.

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Though the material in the book isn't related to calculus, Richard Hammock's Book of Proof is a nice introduction to proofs that goes over direct, contrapositive, contradiction, induction and set containment proof techniques. The problems throughout the book generally consists of proving basic number theory results, so there aren't any extensive prerequisites that are necessary. The books content, as well solutions, can be found on the website http://www.people.vcu.edu/~rhammack/BookOfProof/

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"A Logical approach to discrete math" teachs predicate and propositional calculus and even gives a "formal" epsilon-delta example, in chapter 9, in complete detail with reference to the quantifiers rules applied.

It is a very good book. I have used it in my undergraduate math classes and even consulted it in graduate school.

There is also a proof-checker, for the book, that will look at your proofs and inform you if they are valid or not. It's called CalcCheck and is maintained by one wolfram kahl.

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