Inequality iff Difference is Positive
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This definition needs to be completed. In particular: Turn this into a definition page -- this is how e.g. Smith defines the inequality relation. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Theorem
Let $x, y \in \R$.
Then the following are equivalent:
- $(1): \quad x < y$
- $(2): \quad y - x > 0$
Proof
\(\ds x < y\) | \(\leadstoandfrom\) | \(\ds y > x\) | Definition of Dual Ordering | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds y + \paren {-x} > x + \paren {-x}\) | Real Number Ordering is Compatible with Addition | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds y + \paren {-x} > 0\) | Real Number Axiom $\R \text A4$: Inverses for Addition | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds y - x > 0\) | Definition of Field Subtraction |
Hence the result.
$\blacksquare$
Note
This page or section has statements made on it that ought to be extracted and proved in a Theorem page. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page. |
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