I have a problem with writing a decent proof on upper and lower bounds

of two sets - $\displaystyle A={\frac{n-k^{2}}{n^{2}+k^{3}}$ , $\displaystyle n,k \in \mathbb{N}} $ and $\displaystyle B={\frac{m^{2}-n}{m^{2}+n^{2}} $, $\displaystyle 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 $\displaystyle \epsilon>0 $ and

$\displaystyle \frac{m^{2}-n}{m^{2}+n^{2}} >\epsilon $ and I want to show that for all $\displaystyle n>n_\epsilon $. How can I do it?