Summary -- how to prove a theorem

1. Identify the assumptions and goals of the theorem.
2. Understand the implications of each of the assumptions made. Translate them into mathematical definitions if you can.
3. Either try to massage the definitions and theorems that you identified in (2) into the statement you are trying to prove, or, if that fails
4. Make an assumption about what you are trying to prove and show that it leads to a proof or a contradiction.

The last two items are the only two possible ways to convert your assumptions into proof. These and other possible techniques for proving theorems will be discussed in more detail in the next section.