Reciprocal Function is Strictly Decreasing

Theorem
The reciprocal function:


 * $\operatorname{recip}:\R \setminus \left\{ {0} \right\} \to \R$, $x \mapsto \dfrac 1 x$

is strictly decreasing:


 * on the open interval $\left ({0 \,.\,.\, +\infty} \right)$


 * on the open interval $\left ({-\infty \,.\,.\, 0} \right)$

Also see

 * Reciprocal Sequence Strictly Decreasing
 * Sum of Reciprocals is Divergent: Proof 2
 * Existence of Euler-Mascheroni Constant