# Inequality iff Difference is Positive

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## Theorem

Let $x, y \in \R$.

Then the following are equivalent:

- $(1): \quad x < y$
- $(2): \quad y - x > 0$

## Proof

\(\displaystyle x < y\) | \(\leadstoandfrom\) | \(\displaystyle y > x\) | Definition of Dual Ordering | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle y + \paren {-x} > x + \paren {-x}\) | Real Number Ordering is Compatible with Addition | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle y + \paren {-x} > 0\) | Real Number Axioms: $\R \text A 4$: Inverse Elements | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle y - x > 0\) | Definition of Subtraction |

Hence the result.

$\blacksquare$

## Note

If the notion of an ordering on $\R$ has not already been defined rigorously, this is often taken to be the definition of $x < y$.

## Sources

- 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.3$: Inequalities