- Content—the logic of the proof
- A proof should contain a clear and rigorous chain of
reasoning leading from the hypothesis to the conclusion. You should give
arguments supporting all the steps. (More precisely, give arguments for all
the steps except for those which are ``obvious''. Learning what this
means—i.e., learning how much to write down for each argument—is part of
learning the art of writing proofs). Everything written down should be
relevant to this chain of reasoning: don't start by writing down a list of
things you know, and don't digress as you go along.
- The most common error that students make is to work backward
from the desired conclusion to the given hypotheses. This may be
helpful at the preliminary stage of figuring out how to construct a proof,
but is never correct for the final version; you must rewrite the proof,
moving from what you know to what you want to establish, to be sure that
the logic works in that direction. Another way to say this is that you
should never write down a statement or equality unless you know that
it is true—from definitions or previously established results, from the
hypotheses which are given, or from some chain of reasoning based on
these—or unless you state explicitly that it has not been
established (for example, you might remark that ``We must show that . . .
,'' or a proof by contradiction might begin ``We proceed by contradiction;
suppose then that . . . .'')
- When you use a result from the book or class, say so explicitly, and
indicate that you are aware of and have checked the hypotheses. For
example, ``Since f and g are continuous, Lemma 6.66 implies
that . . . .'' Of course, sometimes verifying the hypotheses involves a
lot of explicit work, after which you might say ``Hence by a theorem proved
in class, . . . .''
- When you are done, read your proof critically. Pretend
that it was written by a stranger, and that you did not know what he or she
was thinking. Does the proof then convince you absolutely? If not, try
- Style—the language of the proof
- Write in complete, grammatically proper sentences.
Remember that the equal sign is a verb. Avoid dangling modifiers.
- Study proofs in the text, in other books, and from the
lectures, to get a feeling for good mathematical style.
- Don't use the notations ∀ and ∃. Don't
introduce mysterious abbreviations to save writing out words.
- Writing up homework
- Don't turn in scratch work. Once you have decided how to
construct a proof, write out the details neatly to turn in.
- Follow up
- When your work is returned to you, read the comments
and be sure you understand their point. If you don't, come to see me to
talk about them.