P-Norm of Real Sequence is Strictly Decreasing Function of P

Theorem
Let $p \ge 1$ be a real number.

Let $\ell^p$ denote the $p$-sequence space.

Let $\mathbf x = \sequence {x_n} \in \ell^p$.

Suppose $\mathbf x$ is not a sequence of zero elements.

Let $\norm {\mathbf x}_p$ denote the $p$-norm.

Then the mapping $p \to \norm {\mathbf x}_p$ is strictly decreasing $p$.

Proof
By derivative of $p$-norm $p$:

By $p$-Norm is Norm, $\norm {\mathbf x}_p > 0$ for $\mathbf x \ne \sequence 0$.

By previously derived inequality, the term in parenthesis is negative.

Hence:


 * $\forall p \ge 1: \dfrac \d {\d p} \norm {\mathbf x}_p < 0$