Please help me to find (if they exist): the least upper bound, greatest lower bound, the smallest and the largest elements of these sets. $$\left\{ \frac{p-q}{p+q},\space p \text{ and } q\in \mathbb{N},\space p>q\right\}$$ and $$\{x\in \mathbb{Q},\space x^2\leq3 \}$$
[Math] Find the least upper and greatest lower bound of these sets
real-analysis
Best Answer
A way to approach the first problem is in two cases, $q=0$ and $q\ne0$. If $q=0$ then $\frac{p-q}{p+q} = 1$. If $q\ne0$ then $0 \lt \frac{p-q}{p+q} \lt 1$.
A way to be sure that $\frac{p-q}{p+q} \gt 0$ is to consider when $p$ and $q$ are the closest they can be, $1$ apart. i.e. $p = q +1$. Then $\frac{p-q}{p+q} = \frac{1}{2q+1} $ and $\lim_{q \to \infty} \frac{1}{2q+1} = 0 $
So an upper bound would be $1$ and a lower bound $0$. The set contains $1$, $1$ is also the upper bound, so $1$ is the largest element of the set. The lower bound is $0$, but $\frac{p-q}{p+q}$ can never be $0$, so the set does not have a smallest element.
note: If you don't take $0 \in \mathbb{N}$ then there is no largest element, because $\frac{p-q}{p+q}\ne 1$.