I have previously written about the Cauchy-Schwarz Inequality.

Here is another proof of this fun inequality. We start at a similar point to the previous proof by applying the fact that the square of any number is always non-negative. Suppose is an arbitrary real number, then for

.

This would be true for whatever real numbers we used, but there is a reason for using these particular real numbers in this particular way. This is because we can construct a very useful sum,

,

which with some rearrangement can be seen to be a polynomial in ,

This is a quadratic equation. For a quadratic expression,

,

the minimum value over all possible real values of occurs when

.

If we apply this information to the quadratic expression, which we have previously constructed, we get that the minimum should occur at

.

If we substitute this value into the quadratic expression, we further see that

,

and with some algebraic manipulation that,

This is the Cauchy-Schwarz Inequality. However, it may bother some readers that it does not look exactly like the form of the Cauchy-Schwarz equality that I have given. If so, note that if we take the square root of both sides,

But,

and so,

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