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.

1. Assume that $\emptyset\neq A\subseteq B$ and that $B$ is bounded from above. Show that $A$ is bounded from above, and moreover that $\sup A\leq \sup B$.
2. 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.