Find $\inf$ and $\sup$

Question

Find $\inf$ and $\sup$ of $A=\left\{ \dfrac{2013}{1+\epsilon+\epsilon^{-1}}: \epsilon \in (0,1)\right\}$ . Check if $A$ has the biggest element and the smallest element.

Answer 1

$$ϵ∈(0,1)\rightarrowϵ+ϵ^{-1}> 2\\1+ϵ+ϵ^{-1}>3\\0<\frac{1}{1+ϵ+ϵ^{-1}}< \frac{1}{3}\\0<\frac{2013}{1+ϵ+ϵ^{-1}}< \frac{2013}{3}\\$$

Answer 2

You can prove that the function $$ f(x) = x + \frac1x $$ is decreasing on $(0,1/2)$, then increasing on $(1/2,1)$.