Solutions to WPR I Sample Exam

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# 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.