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

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

Let ${\ell^p}_\R$ denote the real $p$-sequence space.

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

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

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

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

Proof
Equality holds only for a sequence of zero elements.

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

Then:

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

By $\paren \star$, the term in parenthesis is negative.

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

Hence:


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