Minkowski's Inequality for Sums/Index 2

Theorem
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \ge 0$ be non-negative real numbers.

Then:
 * $\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^2}\right)^{1/2} \le \left({\sum_{k \mathop = 1}^n a_k^2}\right)^{1/2} + \left({\sum_{k \mathop = 1}^n b_k^2}\right)^{1/2}$

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

Also see
This result is a special case of Minkowski's Inequality for Sums, where $p = 2$.