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Section 9.5 Additional Problems
Problem 9.5.1 .
Mimic the definitions of an upper bound of a set and the least upper bound (supremum) of a set to give definitions for a lower bound of a set and the greatest lower bound (infimum) of a set.
Note : The infimum of a set \(S\) is denoted by \(\inf(S)\text{.}\)
Problem 9.5.2 .
Find the least upper bound (supremum) and greatest lower bound (infimum) of the following sets of real numbers, if they exist. (If one does not exist then say so.)
\(\displaystyle S=\left\{\frac{1}{n}\,|\,n=1,2,3,\ldots\right\}\)
\(T=\left\{r\,|\,r\right.\) is rational and \(\left.r^2\lt 2\right\}\)
\(\displaystyle (-\infty,0)\cup(1,\infty)\)
\(\displaystyle R=\left\{\frac{(-1)^n}{n}\,|\,n=1,2,3,\ldots\right\}\)
\(\displaystyle (2,3\pi]\cap\QQ\)
The empty set \(\emptyset\)
Problem 9.5.3 .
Let \(S\subseteq\RR\) and let \(T=\{-x|\,x\in S\}\text{.}\)
Prove that \(b\) is an upper bound of \(S\) if and only if \(-b\) is a lower bound of \(T\text{.}\)
Prove that \(b=\sup S\) if and only if \(-b=\inf T\text{.}\)
