I recently read The Cauchy-Schwarz Master Class by Michael Steele which, in addition to teaching the reader about inequalities, has a lot of fun and varied things to say about the Cauchy-Schwarz inequality.
First things, first, the Cauchy-Schwarz inequality in it’s simplest form says the following:
This holds for any pair of lists of numbers, and , for . We know that the square of any number is greater than or equal to zero. Thus, for two real numbers,
but with some algebra,
We can easily apply this knowledge to the lists and for .
Here we apply a rather entertaining trick. Suppose, we had two lists of numbers which we defined using the original lists as follows:
This would change the situation in this way,
So, it follows that
This concludes the proof of the Cauchy-Schwarz inequality.