upper and lower bounds of a set with two variables

**Lisa91** · November 11th 2012, 05:05 PM

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?

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