I'm having trouble understanding proofs. I don't know how to come up the answers on my own. I search through the book looking for the answer. I understand what they are doing, but I don't know how to do it.
First you should think about why you want a proof. You want a proof so you are SURE the statement is true. Put in terms of experience, you want to be sure that the next example you see will not be a COUNTEREXAMPLE which demonstrates the sentence (or argument) is false.
The first thing I would do, as a mathematician, is to look at one or two examples. This is to make sure I know what all the words mean. It also starts me thinking about any connections which keep coming up in these examples.
A next step might be to look for a counterexample. What would happen if the sentence (or argument) is false. When I see why I can't find a counterexample, and can express this - then I have a proof. (formally, this is called a proof by contradiction).
Remember the guts of a proof is a convincing argument (convincing to YOU and your teacher) that what the sentence says always happens. The fact that there are standard forms for some proofs, should not dominate your search. Of course, familiar examples of previous proofs can suggest approaches to try.
Finding proofs is like any problem solving. It is good to have several approaches to try, and to know when to switch strategies!
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