Reciprocal of Real Number is Non-Zero

Theorem

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

Proof

Aiming for a contradiction, suppose that:

$\exists x \in \R_{\ne 0}: \dfrac 1 x = 0$

From Real Zero is Zero Element

$\dfrac 1 x \times x = 0$

But from Real Number Axiom $\R \text M4$: Inverses for Multiplication:

$\dfrac 1 x \times x = 1$

The result follows by Proof by Contradiction.

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(p)}$