However, the thoughts which a mathematician has when trying to find a proof (in step 1) are often quite different from what is written down in step 2. The aim of step 2 ist to conserve your thoughts for further people (or a later version of yourself), such that the reader can understand the proof investing as little time / effort as possible. What you can read in most math books is the result of the second step. Then, if one has a solution, one tries to write it down in a short and elegant way. Usually, this job splits into two steps: At first, one tries some brainstorming (with a pencil on a piece of paper), trying to find a way to prove convergence or divergence. In this chapter, we will explain how convergence and divergence of a sequence can be proven.
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