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 »