Supremum/Maximum and Infimum/minimum of a given set Determine $\sup E$, $\inf E$, and (where possible) $\max E$, $\min E$ for the set $E = \{ \sqrt[n]{n}: n \in \mathbb{N}\}$.
Attempt: I've written that $\inf E = 1 = \min E$.
When it comes to finding $\sup E$, I've noticed punching in increasing values of n on my calculator, the elements of $E$ seem to never go past about $1.4\ldots$, but I still don't know what $\sup E$ is. How do I figure this out?

The function $f(x)=x^{\frac{1}{x}}$ has derivative
$$ f^{\prime}(x)=x^{\frac{1}{x}}\frac{1-\log x}{x^2}$$
Therefore $f$ has its global maximum on $[1,\infty)$ at $x=e$, and is increasing on $[1,e)$ and decreasing on $(e,\infty)$. Therefore the only values of $n$ you need to check are $n=2$ and $n=3$. And $3^{\frac{1}{3}}>\sqrt{2}$.
