4.8 Summary and final words of encouragement

In this chapter, we discussed common proof techniques through examples.

As a final comment, it is important that you understand something about the way proofs are typically presented. Proofs are usually written concisely to make following the argument as clear as possible to the reader; but this style hides most of the work that has gone into finding the proof. Often, most of the effort involved in proving a claim is hidden when a proof is written out.

When you are trying to write proofs, you will likely have to do a lot of rough work (playing with examples and so on) before you find a proof. Struggling with difficult ideas is an essential part of doing mathematics, so you need to embrace this challenge.

All mathematicians find mathematics challenging!