Sandbox
Problem 1. We say that $m$ is the maximum of a set $A$ if $m\in A$ and $m\geq a$ for all $a\in A$.
Suppose a set $A$ of real numbers has a maximum, call it $m$. Show that $m$ is also the supremum of $A$.
Problem 2. Let $A=\{x\in\mathbb{Q}\ |\ x^2\leq 3\}$. Show that A is bounded from above, but that $A$ has no maximum.
Problem 3. Let $A$ and $B$ be two sets of real numbers.
- Assume that $\emptyset\neq A\subseteq B$ and that $B$ is bounded from above. Show that $A$ is bounded from above, and moreover that $\sup A\leq \sup B$.
- State a similar theorem for sets that are bounded from below. (No proof required.)
Problem 4.
A set is called bounded if it is bounded from above and bounded from below. Assume that every non-empty bounded set of real numbers has both an infimum and a supremum. Show that this implies the Completeness Axiom for the Real Numbers.
Hint: This is not as easy as it may look at first glance.
<script type="text/javascript" src="http://www.wolfram.com/cdf-player/plugin/v2.1/cdfplugin.js"></script> <script type="text/javascript"> var cdf = new cdfplugin(); cdf.embed('http://helmut.knaust.info/NB/Bungee.cdf', 777, 943); </script>
