For the following subset of the rational numbers, determine if the set 1) has a maximum 2) an upper bound 3) a least upper bound.
The set is: $\{ \frac {a}{b} \space | \space a,b \in \mathbb {Z^{+}}, \space a^2 -ab - b^2 <0 \}$.
In this definition, $0$ isn't in $ \mathbb {Z^{+}} $. It also has a note: "you may assume there is no rational number $r$ with $ r^2 =5$"
I don't understand the note, mostly because I don't see how this set is bounded in any way.
Is this set somehow bounded? If it is what is its least upper bound?
Hint: Observe \begin{align} a^2-ab-b^2= \left(a-\frac{b}{2}\right)^2-\frac{5}{4}b^2 = b^2\left\{ \left( \frac{a}{b}-\frac{1}{2}\right)^2-\frac{5}{4}\right\}<0 \end{align} which means \begin{align} \left( \frac{a}{b}-\frac{1}{2}\right)^2<\frac{5}{4}. \end{align}