Question
Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that$$\inf A=-\sup (-A)$$
Step 1
Since $A$ is nonempty and bounded below, it has an infimum, say $\inf A = a$. This means that $a$ is the greatest lower bound of $A$. In other words, $a \le x$ for all $x \in A$, and if $b$ is any lower bound of $A$, then $b \le a$. Show more…
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