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I am mid-way through the second iteration of working problems in "The Book of Proof" by Richard Hammack. It is book on how to learn how to do mathematical proofs. Despite the review, I find that I am still having difficulty doing some of the more difficult problems involving direct proofs, contra-positive proofs, and proofs by contradiction. I believe now that the reason for this is that I have trouble doing proofs backward as well as forward (or meeting in the middle). Where can I find more illustrative examples and/or solved problems to learn this skill better (step by step) so it comes more automatically? I feel like it is key to my success learning how to do proofs and I am still struggling with it.

mont2223
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  • Unsure if this comment is on point. Generally, when I try to construct an $\epsilon,\delta$ proof that (for example) $~\displaystyle \lim_{x\to a} f(x) = L~$, I derive a candidate relationship between $\epsilon$ and $\delta$. Then, working backwards, I retrace my steps to verify that the candidate relationship does allow the assertion that $0 < |x-a| < \delta \implies |f(x) - L| < \epsilon.$ Therefore, virtually all my work on these types of problems involves a backwards-working derivation, followed by a forwards looking proof. – user2661923 Feb 20 '22 at 04:57
  • Alternatively, when I have an assertion that I suspect is true that I am having trouble solving, I write the premise at the top of the page (e.g. Step 1) and the conclusion (e.g. Step 10) at the bottom of the page. Then, I try to simplify Step 10, by considering what Step 9 assertion would imply Step 10. Similarly, I try to find a Step 2 that is implied by Step 1. Then, I simultaneously try to work forward from Step 2 and backwards from Step 9. Note that (for example) most but not all algebraic manipulations are bi-conditional. see next comment. – user2661923 Feb 20 '22 at 05:00
  • That is, if Step 9 implies Step 8, then Step 8 usually also implies Step 9. One common type of exception is where you square both sides. That is $(-2)^2 = (2)^2$ does not imply that $(-2) = (2)$. – user2661923 Feb 20 '22 at 05:02
  • https://www.opepp.org/lesson/hsdm-unit7-tool-for-field/ – DanielWainfleet Feb 20 '22 at 06:49
  • Polya, George – English translation: Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving, 2 volumes, Wiley 1962 (published in one vol. 1981). (Original in German.) – DanielWainfleet Feb 20 '22 at 06:54

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