Minkowski's Inequality

Theorem
Let $$a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$$ be real numbers.

Then $$\left({\sum_{k=1}^n \left({a_k + b_k}\right)^2}\right)^{\frac 1 2} \le \left({\sum_{k=1}^n a_k^2}\right)^{\frac 1 2} + \left({\sum_{k=1}^n b_k^2}\right)^{\frac 1 2}$$.

Proof
$$ $$ $$ $$

The result follows from Order Preserved on Positive Reals by Squaring.