Finding min, max, inf, sup for $\frac{m*n}{4m^2+n^2}; m\in \mathbb{Z}, n \in \mathbb{N}$

Question

What is the proper way to find $\inf, \sup, \max, \min$ for:

$A=\{\frac{m*n}{4m^2+n^2}; m\in \mathbb{Z}, n \in \mathbb{N}\}$

I'm not sure what is the proper proof outline for this case. Thanks!

Answer 1

Check for $m = 0$ and $n = 0$. Then ($n \neq 0 \wedge m \neq 0$) divide numerator and denominator by $mn \neq 0$. $$ A = \left\lbrace \frac{1}{4 \frac{m}{n} + \frac{n}{m}} \left| m \in \mathbb{Z}\smallsetminus \lbrace 0 \rbrace \wedge n \in \mathbb{N}\smallsetminus \lbrace 0 \rbrace\right.\right\rbrace \cup \lbrace 0 \rbrace $$

Now let $q = \frac{m}{n}$. $q \in \mathbb{Q} \smallsetminus \lbrace 0 \rbrace$. So: $$ A = \left\lbrace \frac{1}{4q + \frac{1}{q}}; q \in \mathbb{Q} \smallsetminus \lbrace 0 \rbrace \right\rbrace \cup \ \lbrace 0 \rbrace $$

I have good day, so I made your homework. But think next time, please.