CRN 11982: HW 1

Problem 1. Exercise 1.3.2.

Problem 2. Exercise 1.3.3(a)(b).

Problem 3. Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ and $t$ both are suprema of $A$, then $s=t$. (Suprema are unique.)

Problem 4. A real number $m\in\mathbb{R}$ is called the maximum of the set $A\subseteq\mathbb{R}$, if $m\in A$ and $m\geq a$ for all $a\in A$.

1. Show: If $m$ is the maximum of $A$, then $m$ is also the supremum of $A$.
2. Let $A=\{x\in\mathbb{Q}\ |\ x^2\leq 5\}$. Show that A is bounded from above, but that $A$ has no maximum.

Problem 5. Show that the Nested Interval Property together with the Archimedean Principle implies the Axiom of Completeness.