This geometric proof of irrationality of the Golden Section gives an example of the easiest and most self-evident application of infinite descent.
By definition, the rectangle ABCD is a golden rectangle, if after cutting off the square B’BCC’, the remaining rectangle ADC’B’ is similar to the original one, ABCD. The ration of lengths of sides of a golden rectangle is called the golden ration.
If the golden ratio was rational, a “golden rectangle” could be drawn on square grid paper. After cutting a square from it we get a smaller “golden rectangle” drawn on square grid paper. By principle of infinite descent, this is impossible — hence the golden ratio is irrational.
Hmm, we did not even care about the numeric value of the golden ratio…