# Reciprocal of Strictly Positive Real Number is Strictly Positive

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## 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 \ \$ $\displaystyle x \times \dfrac 1 x$ $<$ $\displaystyle 0 \times 0$ Order of Real Numbers is Dual of Order Multiplied by Negative Number $\displaystyle \leadsto \ \$ $\displaystyle 1$ $<$ $\displaystyle 0$ Real Number Axioms: $\R M4$: Inverse

But from Real Zero is Less than Real One:

$1 > 0$

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