To print this page click the "print" button at the bottom or on the left. In the window that opens, click on all the "+ Show the Proof" lines to display all of the answers. Then print.

This page will be updated with solutions to the sample WPR.

# Problem 1

**Claim:** $\log_{10}(5)$ is irrational.

# Problem 2

**Claim:** $|\vec{x}+\vec{y}|^2\leq|\vec{x}|^2+|\vec{y}|^2$ if and only if $2\langle\vec{x},\vec{y}\rangle\leq0$.

# Problem 3

**Question:** (a) If a function is not continuous at *x*, then there exists $\epsilon>0$ such that…

**Question:** (b) What does this imply about the function $f(x)=1$ for $x\leq0$ and $f(x)=-1$ for $x>0$?

# Problem 4

**Question:** Does the *least upper bound property* hold on the set of integers $\mathbb{Z}$?

# Problem 5

**Question:** Is the subset $\{(x,y)\in\mathbb{R}^2:\min\{|x|,|y|\}\leq 1\}$ convex?

# Problem 6

**Claim:** $GLB(S)\leq LUB(S)$ for any nonempty bounded subset *S* of the real numbers.

# Problem 7

**Claim:** (a) The function $(x,y)\mapsto\max\{|x|,|y|\}$ is a **norm**.

**Claim:** (b) The function $f(x)=| |x\cdot\vec{v}| |$ is continuous.

# Problem 8

**Claim:** Show that for an altered definition of cuts, in which for $A|B$ the set *B* has **no smallest element** rather than the set *A* having **no largest element**, the *least upper bound property* still holds.

## Comments/questions on the first WPR sample exam