The arithmetic-harmonic mean inequality

Cauchy-Schwarz. Apply Cauchy-Schwarz to the vectors $(\sqrt{a_1},\ldots,\sqrt{a_n})$ and $(1/\sqrt{a_1},\ldots,1/\sqrt{a_n})$:
$$\left(\sum_{i=1}^n \sqrt{a_i}\cdot\frac{1}{\sqrt{a_i}}\right)^2\le\left(\sum a_i\right)\left(\sum\frac{1}{a_i}\right).$$
The left side is $n^2$, so
$$n^2\le\left(\sum a_i\right)\left(\sum\frac{1}{a_i}\right).$$

Rearranging.
$$\frac{n}{\sum\frac{1}{a_i}}\le\frac{\sum a_i}{n}.$$

Conclusion. The harmonic mean is at most the arithmetic mean, with equality iff all $a_i$ are equal.