Find the sup and inf of a set with parameter

Question

Find the sup and the inf of this set $$E=\left\{ \frac{m^{\lambda} + n^{\frac{1}{\lambda}}}{m+n}, m,n \in \mathbb{N} \right\}$$ as $\lambda$ changes in $\mathbb{R}^+$.

I'm struggling in showing that the inf is $1$.

Answer 1

For $\lambda = 1$ it holds $$E = \{1\}$$ and we are done. For $\lambda \not= 1$ it holds $$\inf E = 0 \\ \sup E = \infty$$… so no wonder you are struggling.

Assume $\lambda > 1$ then show that $$\lim_{n\to\infty} \frac{m^{\lambda} + n^{\frac{1}{\lambda}}}{m+n} = 0$$ and $$\lim_{m\to\infty} \frac{m^{\lambda} + n^{\frac{1}{\lambda}}}{m+n} = \infty$$

For $0 < \lambda < 1$ substitute $\overline{\lambda} = \frac{1}{\lambda} > 1$ and interchange the roles of $m$ and $n$ to see the results above still hold.