# Cauchy’s Crazy Induction

Standard

Many people know about AM-GM inequality, we have that for any list of $n$ nonnegative real numbers $a_1, a_2, \ldots , a_n$,

$\sum_{k=1}^{n} a_k \geq n \sqrt[n]{\prod_{k=1}^{n}a_k}$

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

Cauchy, Augustin-Louis (1821). Cours d’analyse [source: https://archive.org/details/coursdanalysede00caucgoog ]

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:

• $P(k) \Rightarrow P(2k)$
• $P(k) \Rightarrow P(k-1)$

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 $n$ . 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 $x_i$ with $0 \leq x_i \leq \frac{1}{2}$ for $i = 1, \ldots, n$ are real numbers, then

$(\prod_{i=1}^n \frac{x_i}{1-x_i})^{1/n}\leq \frac{\frac{1}{n}\sum_{i=1}^n x_i}{1-\frac{1}{n}\sum_{i=1}^n x_i}$

2) Cauchy’s Inequality or  Lagrange’s inequality :  Let  $n\geq 1$ and $x_1,\ldots, x_n,y_1, \ldots, y_n$ be a set of $2n$ real numbers. Then

$(\sum_{i=1}^n x_i y_i)^2 \leq (\sum_{i=1}^n x_i^2)(\sum_{i=1}^n y_i^2)$

References: