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,

and

;

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

.

In summary,

.

This concludes the proof of the Cauchy-Schwarz inequality.

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