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I am planning on being a math major this coming fall and will likely be enrolled in honors calculus. In preparation for this course and having found myself with an abundance of free time, I have begun working through Spivak's calculus to get out ahead of the curve.

Unfortunately, I have found that my proof writing ability, my knowledge/intuition of mathematic logic, and my understanding of the requisite set theory to write these proofs is not at the level of Spivak's questions. What should I do to bring myself up to speed (books, resources etc.)? For reference, I have solid knowledge of computational single and multi variable calculus, but little to no experience writing formal proofs besides a super dumbed-down delta-epsilon proof.

Please delete if this breaks any rules, as this is my first post here.

Andrew Benedict
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    you can always start answering questions on this site to practice writing. – Asinomás Mar 24 '21 at 16:28
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    I hope you'll get some good guidance here; but as someone teaching an honours calculus course right now, I want to assure you that you don't need to have already mastered those skills going in! Part of the course is giving you a chance to practice those skills. – Greg Martin Mar 24 '21 at 16:38
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    I agree with Greg Martin. Spivak is known for having many challenging problems. Don't be discouraged if you can't "ace" them. You will get a lot from them, even if you have to look up the answers. Working your way through will help you begin to build up your proof reading/writing skills. – Ben Mar 24 '21 at 16:50
  • If you haven't already, you should pick up a copy of the Spivak Answer Book. Very helpful for self-study – Ben Mar 24 '21 at 16:53
  • And just keep struggling. It’s not unusual at first to need a few days of struggle just to do one problem correctly. You’ll get better and faster as you get the hang of it. – Deane Mar 24 '21 at 17:38
  • Google: software to learn the basic methods of proof – Dan Christensen Mar 24 '21 at 21:13
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    You might look at the books listed in this question. In particular, Velleman's "How to Prove It" is recommended in Peter Smith's answer and Hammack's "Book of Proof" is freely available online, as are a couple other books under "Introduction to Proofs" at AIM's list of approved open texts. – Mark S. Mar 28 '21 at 19:45

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Do you know Daniel Velleman's fine book How To Prove It (CUP)? This could well be worth looking at as a first port of call. From his blurb:

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching 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. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets.

Many students have found this book of great help.

Peter Smith
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  • You might also look at Velleman, Calculus: A Rigorous First Course, which gives a rigorous presentation of calculus but is not as challenging as Spivak. – Dan Velleman Mar 30 '21 at 19:37