How to write proofs: a quick guide

This is a 17 page pamphlet aimed at mathematics students who are perplexed about how to write proofs, having never written proper mathematical proofs before.  It's chatty, with examples of good proofs and bad proofs.  Here are the contents:

  1. What does a proof look like?
  2. Why is writing a proof hard?
  3. What sort of things do we try and prove?
  4. The general shape of a proof
  5. What doesn't a proof look like -- popular ways to write a bad proof
  6. Practicalities: how to think up a proof
  7. Some more specific shapes of proofs
  8. Proof by contradiction
  9. Exercises: What is wrong with the following "proofs"?

I wrote it for an evening "workshop" I ran for my calculus students at the University of Chicago, so the examples are taken from the beginning of that course: field axioms, functions etc.  All the examples of erroneous proofs come from recurring problems I've seen from students in both Cambridge and Chicago.  The aim of this pamphlet is to help iron out those problems.

I would be interested to hear your comments about it, whether you're a student or faculty member.  You're welcome to distribute it to your students - in which case I'd be particularly interested to hear if it's useful.

You can download it here:


And my e-mail address is: e {dot} cheng {at} sheffield {dot} ac {dot} uk

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