Find $\sup{A},\inf{A},\max{A},\min{A}$ where: $$A=\left\lbrace\frac{2013}{1+\epsilon+\epsilon^{-1}}:\epsilon\in(0,1)\right\rbrace$$
I suspect that $\sup{A}=\frac{2013}{3}, \inf{A}=0$ and max and min don't exist, I can easily prove that my candidates are upper and lower bounds, but how to proceed from there?
Your answer is correct. You can prove it by using that $A$ is strictly increasing in $\epsilon$, in the interval $(0,1)$ since $$\frac{∂A}{∂\epsilon}=−2013\frac{\epsilon^2-1}{\epsilon^4+2\epsilon^3+3\epsilon^2+2\epsilon+1}>0, \qquad \forall\epsilon\in(0,1)$$