# Math Help - infimum question

1. ## infimum question

let A be a nonempty subset of positive real numbers with
$
\inf \ A \neq\ 0 .
\text{ prove that } \sup \{ \frac{1}{a} \ ; a \ \in \ A \} \ = \frac {1}{\inf A}$

2. Originally Posted by flower3
let A be a nonempty subset of positive real numbers with
$
\inf \ A \neq\ 0 .
\text{ prove that } \sup \{ \frac{1}{a} \ ; a \ \in \ A \} \ = \frac {1}{\inf A}$
Suppose that $0<\lambda =\inf(A)$. If $a\in A$ then $\frac{1}{a}\le \frac{1}{\lambda}$.
Therefore the set $\left\{\frac{1}{a}:a\in A\right\}$ has a $\sup$ $\gamma\le \frac{1}{\lambda}$.

Suppose that $\gamma <\frac{1}{\lambda}$. Show that leads to a contradiction.