Proofs that involve Tricks

Question

When I am doing a homework problem or self-studying, I can usually do proofs that only require theorems or definitions. But lately I've been trying to read more advanced texts in analysis and algebra on my own and its clear to me that I cannot solve problems just by using the definitions or theorems. For example:

Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed.

In fact after spending weeks on this problem, I decided to look up the solution. Although I understand the solution, I don't think I could have ever come up with such a solution to the problem.

Is struggling like this for weeks the best way to solve more difficult problems? If not, what should I do differently?

Answer 1

Surely, it is better to spend some time thinking about a theorem or a lemma before reading the proof directly. Otherwise, you might not appreciate the value of the proof. Moreover, you might be able to come up with the proof before reading the solution, maybe using a different approach.

However, if you try to prove something and you keep using the same approach and failing. Then this might be the time to do something different. In addition, if you are learning and you get stuck at a proof for weeks then you are missing the oppurtunity to learn something new during this time. I usually try to prove theorems of a math book before looking at the proof given in the book. On average, I allow about one hour after which if I still fail, I look at the book's proof.

A nice book about problem solving skills (it is mostly about high school math though) is the "Art and Craft of Problem solving"

Answer 2

I think it's always a good idea to struggle with a problem for a while and avoid looking up the answer. Even better, though, is to try and attack it from many different angles. If you're stuck, don't keep trying to hit it with the same hammer - go look for different tools. I've had to come back to difficult problems sometimes a couple years after first encountering them because I didn't have the right understanding yet. Or sometimes all it takes is reading a different book on the same subject - it might be able to provide insights that work better with your way of seeing things.

As for the problem of looking at a solution and feeling like it was "just a trick", I think this happens to a lot of us, especially when encountering new material. Don't feel bad, eventually you'll realize where the trick comes from and it'll make the proof seem all that more elegant. The best thing to do is to keep going back to old problems with new eyes - you'll realize that some of the things that felt like tricks 3 months or 3 years ago now seem perfectly natural. Eventually you'll be the one making up the tricks!

Answer 3

Two references that might help you out:

How to Prove It: A Structured Approach by Daniel Velleman.

• This you may want to have, perhaps as a reference.

Thinking Mathematically (2nd Edition), by J. Mason, L. Burton, and K. Stacey.

Thinking Mathematically unfolds the processes which lie at the heart of mathematics. It demonstrates how to encourage, develop, and foster the processes which seem to come naturally to mathematicians. In this way, a deep seated awareness of the nature of mathematical thinking can grow. The book is increasingly used to provide students at a tertiary level with some experience of mathematical thinking processes.

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Struggling with a problem or a proof isn't a bad thing. Nothing truly innovative is proven overnight: ideas, insight, strategies, ... often need time to "ferment" to grow to fruition. But you needn't torture yourself. Spend time both writing AND reading proofs. The more exposure you have to different styles of exposition, different approaches to proofs, and various "crafty" techniques, the more readily you'll see when and how such approaches and/or techniques might apply to problems YOU want to prove.

We learn by doing, and we learn by observing (reading) and modeling what others have done. Each has its place. Better yet, combine the two, and you'll be on your way!

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Answer 4

I recommend the book by Terence Tao. Awarded the Fields Medal in 1996. Most honor that a mathematician might have.

Solving mathematical problems, a personal perspective.