What's the supremum and infimum of the set $M=\left \{ \left ( -\frac{1}{3} \right )^{m}-\frac{5}{n}\mid m,n \in \mathbb{N} \right \}$
Are they included in the set?
$$\inf = \left(-\frac{1}{3}\right)^{1}-\frac{5}{1} = -\frac{16}{3}$$
It's obviously included in the set for $m=1$ and $n=1$ which also means that the infimum is a minimum.
$$\sup=\lim_{m,n\rightarrow \infty }\left( \left ( -\frac{1}{3} \right )^{m}-\frac{5}{n} \right )=0$$
The supremum isn't in the set.
I have my doubts how I solved this... Is it correct?
To make $\left( -\tfrac 1 3 \right)^m - \tfrac 5 n$ as large as possible, you need $m$ to be even so that the negative sign is killed and also as small as possible since as $m$ gets large $\left( \tfrac 1 3 \right)^m$ gets small. Thus we should take $m=2$. However, to ensure that we aren't subtracting off too much, we would like $n$ to be as large as possible. Indeed, setting $m=2$ and taking $n\to \infty$ gives that the supremum is $\tfrac 1 9$. The supremum is not a member of the set in this case since it required taking $n\to\ \infty$ (i.e., it is never actually achieved for finite $n$).