Exercise
26
[20Y](Proposed on 2022-12) Calculate supremum and infimum of the following sets (where \(n,m\) are integers).
\begin{eqnarray*} \left\{ \frac{mn}{m^ 2+n^ 2}: n,m≥ 1 \right\} \quad ,\quad \left\{ \frac{mn}{m+n}: n,m≥ 1 \right\} \\ \left\{ 2^ n+2^ m : n,m∈ℕ \right\} \quad ,\quad \left\{ 2^ n+2^ m: n,m∈ℤ \right\} \\ \left\{ \frac{m^ 2-2} n : n,m∈ℤ,n≠ 0 \right\} \quad ,\quad \left\{ \frac{m+1}{m^ 2} : m∈ℤ,m≠ 0 \right\} \end{eqnarray*}