How to find infimum and supremum

Question

I have to find the infimum and supremum of the set $A = \left\{\frac{n + k^2}{2^n + k^2 + 1} : n,k \in \mathbb{N}\right\}$. We assume $0 \notin \mathbb{N}$. $\inf A = 0$ because $\lim_{n \to \infty}{\frac{n + k^2}{2^n + k^2 + 1}} = 0$ but I don't know how to find the supremum. Moreover, is there any general method of finding infima and suprema of sets like $A$?

Answer 1

When $n$ is very large the fraction goes to $0$. Use the same idea for $k$: when $k$ is very large and $n$ is fixed the fraction goes to $1$. So any upper bound must be at least $1$. With this in mind, try to check that $1$ actually is an upper bound. As a hint, note that $2^n+1 > n$.