Minkowski's Inequality for Sums/Index Greater than 1/Proof 2

Proof
Let $\mathbf a$ and $\mathbf b$ be real finite sequences:

Let $\norm {\mathbf a}_p$ denote the $p$-norm of $\mathbf a$:
 * $\norm {\mathbf a}_p := \paren {\ds \sum_{k \mathop = 1}^n {a_k}^p}^{1 / p}$

Define:
 * $q = \dfrac p {p - 1}$

Then:
 * $\dfrac 1 p + \dfrac 1 q = \dfrac 1 p + \dfrac {p - 1} p = 1$

Then: