find the supremum and infimum of $B=\left\{\frac{m n}{3 m^{2}+n}: m \in \mathbb{Z}, n \in \mathbb{N}\right\}$

Question

I am trying to find the supremum and infimum of the following set $B$ just by definition of sup and inf (not limits or derivative, etc.) $$B=\left \{\frac{m n}{3 m^{2}+n}: m \in \mathbb{Z}, n \in \mathbb{N}\right\}$$

What I was trying to do is $$0\leq(m-n)^2=m^2-2mn+n^2 $$ $$ \frac{2mn}{m^2 +n^2}\leq 1$$ which is not helping me so much.
I would like to ask for any advice/guidline how to approach this problem.

Answer 1

The set is not bounded. To prove this, take $n=3m^2.$ Then $$\frac{mn}{3m^2+n}=\frac{m}{2}$$ Since $m\in \mathbb{Z},$ there is no supremum neither infimum.