# Reciprocal of Strictly Positive Real Number is Strictly Positive

## 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:

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

But from Real Zero is Less than Real One:

$1 > 0$

$\dfrac 1 x > 0$
$\blacksquare$