Should I be concerned if I cannot solve most exercises in my textbook?

Question

I'm self studying introductory real analysis. Out of the proofs in each lesson section and the proofs in each exercise section, I can usually work out$\frac{1}{4}$ of them on my own. After trying to solve the other proofs, I give up and consult the solutions manual (although I always understand the solution, at the end).

(1) Should I be concerned that I can't solve most proofs by myself?

(2) In upper level math classes (undergraduate and graduate), is it expected that you be capable of solving most proofs by yourself -- or that, by the end of some section, you simply understand every proof, regardless if you could construct it yourself?

Answer 1

"There is no royal road to geometry" (Euclid to Ptolemy) @Muno, but (to paraphrase somehow the above) we can hope there is always a path. You do your effort, you try, you learn the basic methods, and the more experienced you become the better you can approach a problem - don't hurry and don't be harsh on yourself, definately don't blame yourself (it is much better to "blame it on the boogie" (The Jacksons)). Yourself knows better the time needed to adjust. The only thing you have to ask yourself is if you like spending your precious time on math. All the rest will find the way!

Answer 2

Being unable to solve exercises isn't a tragedy. It is a reminder of what tools you are still missing. Here are some tips that might help you along.

In order to solve a given exercise, you

• need to understand (and remember) every relevant definition and be very clear about the notation involved.
• need to make sure that you understand every detail of the given problem.
• should try to draw a connection between the problem at hand and those that you already were able to solve or results that have been presented to you. A huge portion of exercises can be solved by altering a proof that you've already seen in order to overcome some additional difficulties. Maybe you already know the proof under some additional assumptions or you have been presented a slightly weaker result. In these cases, it's helpful to look at the related proof again and pinpoint the exact reason why this wouldn't work for your particular problem. You may be able to replace these steps with ones that work in your scenario.
• should try to envision a rough sketch of a proof first. Having a decent strategy before diving into the technical difficulties can save you a lot of time by ruling out any approaches that fail for obvious reasons. Draw diagrams, try to see why the claim holds for a very specific example and try to see why the provided assumptions are necessary by constructing counterexamples that don't satisfy at least one of those assumptions.
• should try to remember the statements and at least the key ideas of theorems that have been presented to you. This is closely related to point 3 and 4 on the list as these can only be applied if you recognize the underlying pattern. Sometimes even the faintest memory of having seen something similar in "that green book" can be incredibly valuable.
• trust your instincts. Beginning to learn a new field, it's often the case that our intuition fails us. Nonetheless, your intuition is the most powerful tool at your disposal and no amount of failed predictions should stop you from consulting it. There is nothing wrong with guessing what might work and simply calculating its consequences. Maybe your guess is wrong and once you found a counterexample, it's important to note that this approach cannot work (at which point you should reconsult point 2). Maybe it's right, but you seem to be unable to proof it. In any case, trying a new approach allows you to generate new ideas, predict new paths and hopefully leads you on the right track.
• should try to be optimistic. Rule out only those approaches that you know for a fact don't work. It's quite common to get stuck halfway through a proof at some technical detail that you just don't seem to be able to overcome. At this point it might be helpful to back off a little and try a different angle. Being equipped with some partial results and a better understanding of the task allows you to draw connections that you may have missed before.
• ...

Answer 3

Ability to solve exercise problems indicates a good grasp over the fundamental concepts. If you are able to solve $1/4^{th}$ of these problems, its still a decent percentage. In general the exercise problems are in some way, extensions of the theory developed in the section. What you can try is to check where the hypotheses of the results are used in the proof. What happens if I drop one of these. Can the theorem still be proved or somewhere there is a flaw. I tried doing it once for "Cantor's Intersection theorem" in topology and after a little bit struggle, I was able to generate counter examples by dropping each hypothesis. This practice helps.

Answer 4

I too, had the same problem but I changed my mindset and reminded myself of why I love the purest human art form - Mathematics. It's because of its intrinsic beauty.

When I cannot solve a problem, I change my mindset from "I'm so stupid. I can't solve this." to "Wow, this is such an enlightening proof, trick, application, union of two ideas, etc." Just like someone commented before me, I add it to my mathematical toolbox which prepares me further. I don't get angry with myself for not getting an answer anymore than a young film maker would be angry with himself for not making The Godfather. I look at Mathematics as something bigger than myself as a place where I can just admire and learn from elegant ideas. I, no longer, think I have a point to prove to myself or to anybody else so I don't think of mathematics as an arena to prove my intellectual mettle anymore. I think of it as a gallery I visit as an eager spectator. Each visit prepares me to appreciate the next visit better and, on occasion, add something to the gallery.

I also maintain a book of elegant results and proofs that I thought manifested out of nowhere. Not only do I record the question and the answer, but I also note down exactly what I found elegant about it and why I like it. Soon, you'll find yourself admiring proofs and finding links you didn't see before. You don't have to go to very sophisticated waters to find such results. Elementary mathematics is full of them. (Heron's problem, for example.)

Answer 5

Short answer: You should be cautious of the "I understand it once I see it" argument. I've seen far too many students who think that because they think they understand an argument they can then reproduce it. When tested it emerges that they didn't actually understand some key details.

Go back to one of the chapters you did a while ago. Can you do the problems now without consulting the manual? If not, then you probably don't actually understand it.

Also, be aware - some books are much harder than others. If you're reading something that sets the bar very high, then it's a different thing to struggle than if you're reading a book that breaks every problem into many substeps.