Minkowski's inequality

Expansion.
$$\sum(a_i+b_i)^2=\sum a_i^2+2\sum a_ib_i+\sum b_i^2.$$

Cauchy-Schwarz. $\sum a_ib_i\le\sqrt{\sum a_i^2}\sqrt{\sum b_i^2}$.

Bound.
$$\sum(a_i+b_i)^2\le\sum a_i^2+2\sqrt{\sum a_i^2}\sqrt{\sum b_i^2}+\sum b_i^2=\left(\sqrt{\sum a_i^2}+\sqrt{\sum b_i^2}\right)^2.$$

Conclusion. Taking square roots gives the inequality.

Interpretation. This is the triangle inequality for the Euclidean norm on $\mathbb{R}^n$: $\|a+b\|\le\|a\|+\|b\|$.