xviii For students: How to use this book
When you sit down to work at mathematics it is important to have a sizeable
block of time at your disposal during which you will not be interrupted. As you
read, pay especially close attention to definitions. (After all, before you can think
about a mathematical concept you must know what it means.) Read until you
arrive at a result (results are labeled Theorem, Proposition, Example, Problem,
Lemma, etc.). Every result requires justification. The proof of a result may appear
in the body of the text, or it may be left to you as an exercise or a problem.
When you encounter a result, stop and try to prove it. Make a serious attempt.
If a hint appears after the statement of the result, at first do not read it. Do not try
to find the result elsewhere; and do not ask for help. Halmos [Hal67]pointsout,
“To the passive reader, a routine computation and a miracle of ingenuity come with
equal ease, and later, when he must depend on himself, he will find that they went
as easily as they came.” Of course, it is true that sometimes, even after considerable
effort, you will not have discovered a proof. What then?
If a hint is given, and if you have tried seriously but unsuccessfully to derive the
result, then (and only then) should you read the hint. Now try again. Seriously.
What if the hint fails to help, or if there is no hint? If you are stuck on a result
whose proof is labeled Exercise, then consult the online solution. Turning to the
solution should be regarded as a last resort. Even then do not read the whole proof;
read just the first line or two, enough to get you started. Now try to complete the
proof on your own. If you can do a few more steps, fine. If you get stuck again
in midstream, read some more of the proof. Use as little of the online proof as
possible.
What if you are stuck on a result whose proof is a Problem but there is no online
solution? After a really serious attempt to solve the problem, go on. You cannot
bring your mathematical education to a halt because of one refractory problem.
Work on the next result. After a day or two go back and try again. Problems often
solve themselves; frequently, an intractably murky result, after having been allowed
to rest for a few days, will suddenly, and inexplicably become entirely clear. In the
worst case, if repeated attempts fail to produce a solution, you may have to discuss
the problem with someone else—instructor, friend, mother, . . . .
A question that students frequently ask is, “When I’m stuck and I have no idea
at all what to do next, how can I continue to work on a problem?” I know of only
one really good answer. It is advice due to P´olya, “If you can’t solve a problem,
then there is an easier problem you can solve: find it.”
Consider examples. After all, mathematical theorems are usually generaliza-
tions of things that happen in interesting special cases. Try to prove the result in
some concrete cases. If you succeed, try to generalize your argument. Are you stuck
on a theorem about general metric spaces? Try to prove it for Euclidean n-space.
No luck? How about the plane? Can you get the result for the real line? The unit
interval?
Add hypotheses. If you cannot prove the result as stated, can you prove it
under more restrictive assumptions? If you are having no success with a theorem
concerning general matrices, can you prove it for symmetric ones? How about
diagonal ones? What about the 2 × 2case?