Techniques of proof II: Proof by cases
Little by little does the trick.
Aesop, The Crow and the Water Jar
We have already seen that x = y can be proved by showing that x ≤ y and x ≥ y. In other words we have broken the problem into two cases. This is a very common procedure in mathematics. Often we can break down some problem and tackle each case individually utilizing different methods, or even the same methods, in each case.
The famous Four Colour Problem is that given a map (of the geographical kind) we need only four colours to colour ...