Many people know about AM-GM inequality, we have that for any list of nonnegative real numbers ,

While reading about Arithmetic Mean-Geometric Mean Inequality’s proof at Wikipedia, I came across the reference to Augustin Louis Cauchy‘s original proof:

Here Cauchy used a crazy type of induction process, which can be stated as:**Step 1** : [The base case] Show that the statement is true for an integer .

**Step 2:** [The inductive step] This has two parts:

Completing these two steps proves that the statement is true for all positive integers .

It has a very distinctive inductive step, and though it is rarely used, it is a perfect illustration of how flexible induction can be. The first part of the inductive step shows that the statement is true for larger and larger values of . But that leaves a lot of gaps in between. The second part ensures that all the gaps are taken care of.

I became curious to know about other instances when we can use this from of induction. So here are some other theorems which can be proved using **Cauchy’s Crazy Induction **(for proof refer to [2] given at end of this post):

**1)** **Ky Fan inequality:** If with for are real numbers, then

**2) Cauchy’s Inequality or Lagrange’s inequality **: Let and be a set of real numbers. Then

*References:*

[1] Forward-backward induction, The Math Forum @ Drexel