Hölder's Inequality for Sums/Formulation 2/Equality

Theorem
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
 * $\dfrac 1 p + \dfrac 1 q = 1$

Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.

Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.

Hölder's Inequality for Sums states that:
 * $\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$

We have that:
 * $\ds \sum_{k \mathop \in \N} \size {x_k y_k} = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$


 * $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
 * $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$

for some real constant $c$.