Hölder's Inequality for Sums/Finite/Proof

Hölder's Inequality for Finite Sums: Proof
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 $n \in \N_{>0}$ be a non-zero natural number.

Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ and $\sequence {y_k}_{1 \mathop \le k \mathop \le n}$ be finite sequences in $\GF$.

Then:

Proof
Let $\sequence {x_k}_{k \mathop \in \N}$ and $\sequence {y_k}_{k \mathop \in \N}$ be infinite sequences in $\GF$ such that:
 * $\forall m > n: x_m = y_m = 0$

Then we have: