Reciprocal Sequence is Strictly Decreasing

Theorem
The reciprocal sequence:


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

is decreasing.

Corollary

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

is 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 decreasing.

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

Also see

 * Sum of Reciprocals is Divergent/Proof 2
 * Existence of Euler-Mascheroni Constant