AM-HM Inequality

Theorem
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.

Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.

Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$.

Then $A_n \ge H_n$.

Proof
The arithmetic mean of $x_1, x_2, \ldots, x_n$ is defined as:
 * $\displaystyle A_n = \frac 1 n \left({\sum_{k=1}^n x_k}\right)$

The harmonic mean of $x_1, x_2, \ldots, x_n$ is defined as:
 * $\displaystyle \frac 1 H_n = \frac 1 n \left({\sum_{k=1}^n \frac 1 {x_k}}\right)$

As $\forall k \in \left[{1 \,. \, . \, n}\right]: x_k > 0$, we can express all the $x_k$'s as squares:
 * $\forall k \in \left[{1 \, . \, . \, n}\right]: x_k = y_k^2$

without affecting the result.

Thus we have:


 * $\displaystyle A_n = \frac 1 n\left({\sum_{k=1}^n y_k^2}\right)$


 * $\displaystyle \frac 1 H_n = \frac 1 n \left({\sum_{k=1}^n \frac 1 {y_k^2}}\right)$

Now let's see what happens when we multiply them together:

So $\dfrac {A_n} {H_n} \ge 1$ and the result follows.