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