TITLE: How to attack "if true, prove it; if not true, give a counterexample" question?
QUESTION [41 upvotes]: I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if not true, give a counterexample? Yesterday evening I spent four hours looking for a counterexample, and finally figured that the statement was indeed true. I tried so hard to find a counterexample because I was not able to prove the statement at first. Are there any steps to follow in terms of this type of questions? In particular, sometimes a statement does not look like it is true. In this case, how can I quickly figure out whether I should try counterexamples or keep the efforts on proving the statement is true? The core problem is that this type of question is very time-consuming according to my homework experience. I desperately doubt I could figure out a correct answer during a timed exam next week.

REPLY [8 votes]: To the solutions elsewhere, I'd add some rules of thumb in the nature of pursuing a math degree on the under graduate level. 
I found that when given such a problem, a counter example was usually apparent within a few minutes contemplation. If you couldn't find one by then, you probably had something you needed to prove. 
Review your earlier work. Such questions are usually just a mild extension of something previously addressed in examples or homework. 
Perhaps the prof went easy on us, so such problems seemed among the easiest. I think it more likely I had exceptionally good study partners and we were careful to share info instead of copy for the sake of having homework to turn in.
I recommend making it a group activity and make sure you do your homework. Seems like you are doing the latter very well, spending four hours on a problem on your own. Shows a lot of dedication. But I stress that you do not have to struggle alone.