# CRN 11378: HW 1

**Problem 1.** Let $A$ and $B$ be two non-empty sets that are bounded from above. Show: If $\sup A < \sup B$, then $B$ contains an element that is an upper bound of $A$.

**Problem 2.** 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 $s=t$. (This shows that a set can have at most one supremum.)

**Problem 3.** Suppose that every non-empty set that is bounded from above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.

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

- Show: If $m$ is the maximum of $A$, then $m$ is also the supremum of $A$.
- 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*.