Titu's lemma (Cauchy in Engel form)

Cauchy-Schwarz. Apply Cauchy-Schwarz to $(\sqrt{y_1},\ldots,\sqrt{y_n})$ and $(x_1/\sqrt{y_1},\ldots,x_n/\sqrt{y_n})$:
$$\left(\sum_i \sqrt{y_i}\cdot\frac{x_i}{\sqrt{y_i}}\right)^2\le\left(\sum_i y_i\right)\left(\sum_i\frac{x_i^2}{y_i}\right).$$
The left side is $(\sum x_i)^2$.

Rearranging.
$$\frac{(\sum x_i)^2}{\sum y_i}\le\sum_i\frac{x_i^2}{y_i}.$$

Conclusion. Titu's lemma holds, with equality iff the two vectors are proportional, i.e. $x_i/y_i$ is constant.