# Tag Archives: Supremum

## Proof of sup(1/A) = 1/inf(A)

Theorem. Suppose $$A$$ be a nonempty subset of $$\mathbb{R}$$ and bounded below. Suppose further that the greatest lower bound of $$A$$ is positive. Show that $$\sup B = \frac{1}{\inf A}$$ where $$B = \left\{ 1/x ~|~ x\in A \right\} .$$ Proof. Let $$\alpha = \inf (A) ,$$ $$\beta = \frac{1}{\alpha} .$$ We first have to show… Read More »