Ordering of Reciprocals/Proof 1
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Theorem
Let $x, y \in \R$ be real numbers such that $x, y \in \openint 0 \to$ or $x, y \in \openint \gets 0$
Then:
- $x \le y \iff \dfrac 1 y \le \dfrac 1 x$
Proof
By Reciprocal Function is Strictly Decreasing, the reciprocal function is strictly decreasing.
By Mapping from Totally Ordered Set is Dual Order Embedding iff Strictly Decreasing, the reciprocal function is a dual order embedding.
That is:
- $x \le y \iff \dfrac 1 y \le \dfrac 1 x$
$\blacksquare$