Reciprocal Sequence is Strictly Decreasing/Proof 1

Theorem
The reciprocal sequence:


 * $\langle{a}\rangle: \N_{>0} \to \R$: $n \mapsto \dfrac 1 n$

is strictly decreasing.

Proof
Consider:

$f: \R_{>0} \to \R$: $x \mapsto x^{-1}$

From the Power Rule for Derivatives:

Because $-x^{-2} < 0$ for all $x$ considered, from Derivative of Monotone Function, $f$ is strictly decreasing.

As $f$ and $\langle{a}\rangle$ agree for all $n \in \N_{>0}$, from Monotonicity of Real Sequences, $\langle{a}\rangle$ is also strictly decreasing.