The best advice I can offer on how to solve problems and prove theorems is: solve problems and prove theorems; concoct examples for evidence and hints as to general statements; consider special cases; make guesses and decide if your examples support or invalidate them; try to use, or modify and then use, reasoning you have employed or encountered in other situations; if you become stymied, rest and renew your efforts another day; use pencil and paper; keep a record of your thoughts. The more curious you become, the more experience you will acquire and the more you will learn. Think about mathematics, do mathematics, enjoy mathematics!

For a careful development and illustrations of the above suggestions and many more, I recommend that you read G. Pólya's books, How to Solve It, Princeton University Press, 1945 and Mathematics and Plausible Reasoning (especially Vol. I), Princeton University Press, 1954. When all else fails, then in good conscience you may read from the hints and solutions given below; but only read as much of a solution as you need to complete it by yourself.

It is good to remember that problems fall into three classes: can't, think I can, and have. When you have completed your solution—this means you have it written down so that someone who does not yet know the solution and who is also fussy and critical can read it and understand it without having to fill in details you have neglected to describe—then compare the argument in the text with yours.