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:

But from Real Zero is Less than Real One:
 * $1 > 0$

Therefore by Proof by Contradiction:
 * $\dfrac 1 x < 0$

Also see

 * Reciprocal of Strictly Positive Real Number is Strictly Positive