I have this set:
$M=\left\{\frac{1}{n^{2}}-\frac{2}{m}:n,m\in\mathbb{Z}\setminus \left\{0 \right\}\right\}$
How do I prove it's bounded? Additionally, how can I decide the infimum and supremum? I thought an initial attempt at solving it would be to join the two terms like so:
$a_{n}=\frac{1}{n^{2}}-\frac{2}{m}=\frac{m-2n^{2}}{n^{2}m}$
But where to go from there? I have been advised against using limits, introduced in secondary school, but I don't know any other way.
Also, if you have any advise on how to get better at "identifying" solutions (I feel like a lot of it is basically algebra) that would be amazing!