Students face many obstacles when they are trying to learn how to do proofs. I am trying to convince my students that at least SOME aspects of constructing proofs are relatively routine.
One fairly common type of proof that often gives trouble is what I like to call “proof by definition”. I have not invented this term, but many authors use it to mean something rather different. What I mean is that you are asked to prove something, and once you have substituted the definitions into the statements, there is either nothing or very little left to prove. Yet this kind of proof often gives students problems.
- Sometimes they feel they should be able to prove the result based on an intuitive idea of the relevant concept, instead of using the definition given to them.
- Indeed, sometimes students do not know the relevant definitions at all, but don’t realize that this is a major obstacle to proving the result.
- However, sometimes the fact that they have already finished once they have substituted in the definitions leads them to think that they haven’t done anything, and this is a more subtle problem.
For example, using my notion of absorption, on one question sheet I ask them to prove the following.
Let and let be a sequence of real numbers. Then
if and only if the following condition holds:
for all , the open interval absorbs the sequence .
Now, by the time you have substituted in the definitions, the two statements are either exactly or effectively the same, and you have finished. But this does not feel like a proof.
Perhaps the students are right to be uncomfortable with this? Consider the following “question and unsatisfacory answer”.
Question: working in , prove that is not in the interior of .
is not in the interior of because, for all ,
the open interval is not a subset of .
This time, the student has substituted in the definition, and can see that the statement is true, but the proof is NOT finished. Indeed, so far the student has said just a little more than “ is not an interior point because it is not an interior point”.
In this case, more is expected: the student should justify the claim by observing, for example, that is in the set but is not in the set .
So how are students supposed to know whether they have finished once they have substituted the definitions in and can see that the result is now obviously true?
In fact, there can’t be a definitive answer to this. After all, it is hard to argue with the statement (working in )
“Clearly is not an interior point of .”
Yet, this statement would not be acceptable as part of a proof of itself.
I feel that the difference between the two examples here is that, in the first, after substituting in the definitions, the two statements being compared end up EXACTLY the same. So, although it feels as if nothing has happened, nevertheless the proof is complete.
In the second example, after substituting in the definition, you arrive at a statement that looks to be clearly true. But appearances can be deceiving, and the student could ask whether this new claimed fact is really significantly more clearly true than the original statement was.
Meanwhile, I am putting together a large collection of “proof by definition” questions for students to practise on. These are (mostly) not supposed to be at all interesting! The idea is to get the students fluent in substituting in definitions, and then seeing how easy the rest of the proof can be.
See http://www.maths.nott.ac.uk/personal/jff/Teaching/More-Proofs.pdf for the current version. Perhaps there are large collections of similar routine proofs available on the internet? It would surely be worth compiling a big collection as a universal student “practising routine proofs” resource.
October 27 2009