# Thread: upper and lower bounds of a set with two variables

1. ## upper and lower bounds of a set with two variables

I have a problem with writing a decent proof on upper and lower bounds
of two sets - $A={\frac{n-k^{2}}{n^{2}+k^{3}}$ , $n,k \in \mathbb{N}}$ and $B={\frac{m^{2}-n}{m^{2}+n^{2}}$, $n,m \in \mathbb{N}, m>n}$.

I don't know how to cope with these two variables. I want to prove the
upper and lower bounds (for B supremum = 1 and infimum =1/2) using the
definition. I assume that for the supremum of B $\epsilon>0$ and
$\frac{m^{2}-n}{m^{2}+n^{2}} >\epsilon$ and I want to show that for all $n>n_\epsilon$. How can I do it?