# Reciprocal of Strictly Negative Real Number is Strictly Negative

## Theorem

$\forall x \in \R: x < 0 \implies \dfrac 1 x < 0$

## Proof

Let $x < 0$.

Aiming for a contradiction, suppose $\dfrac 1 x > 0$.

Then:

 $\displaystyle x$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle x \times \dfrac 1 x$ $<$ $\displaystyle 0 \times 0$ Real Number Ordering is Compatible with Multiplication: Negative Factor $\displaystyle \leadsto \ \$ $\displaystyle 1$ $<$ $\displaystyle 0$ Real Number Axioms: $\R \text M 4$: Inverse

But from Real Zero is Less than Real One:

$1 > 0$

$\dfrac 1 x < 0$
$\blacksquare$