Reverse Hölder's Inequality for Sums

Theorem
Let $p < 1$ be a non-zero real number, and let $q < 0$ be a real number such that $\displaystyle \frac 1 p - \frac 1 q = 1$.

Suppose that the sequences $\mathbf{x} = \langle{x_n}\rangle$ and $\mathbf{y} = \langle{y_n}\rangle$ in $\C$ (or $\R$) are such that the series
 * $\displaystyle \left( {\sum_{n=1}^\infty \left\vert{x_n}\right\vert^p} \right)^{1/p}$

and
 * $\displaystyle \left( {\sum_{n=1}^\infty \left\vert{y_n}\right\vert^{-q} } \right)^{-1/q}$

are convergent.

Denote these values by $\left\Vert{ \mathbf{x} }\right\Vert_p$ and $\left\Vert{ \mathbf{y} }\right\Vert_{-q}$, respectively.

Here, the notation $\left\Vert{ \mathbf{x} }\right\Vert_p$ does not denote a norm, but is instead just a convenient notation similar to that of the $p$-norm, which is only defined when $p \ge 1$.

Let $\left\Vert{ \mathbf{x}\mathbf{y} }\right\Vert_1$ denote the $1$-norm of $\mathbf{x}\mathbf{y}$, if $\mathbf{x}\mathbf{y}$ is in the Lebesgue space $\ell^1$.

Then $\left\Vert{ \mathbf{x}\mathbf{y} }\right\Vert_1 \ge \left\Vert{ \mathbf{x} }\right\Vert_p \left\Vert{ \mathbf{y} }\right\Vert_{-q}$.

Proof
The proof is the same as that of Hölder's Inequality (Special Case), except that Reverse Young's Inequality for Products is applied instead of Young's Inequality for Products.