## Suprema and infima of concrete sets

Determine the suprema and infima (if any) of the following sets in the real domain. Are these also maxima and minima of these sets?

• #### Variant 1

$$\displaystyle M=\left\{\frac1n,n\in\mathbb N\right\}$$

• #### Variant 2

$$\displaystyle M=\left\{-\frac1n,n\in\mathbb N\right\}$$

• #### Variant 3

$$M=\{0.3; 0.33; 0.333; 0.3333; …\}$$

• #### Variant 4

$$\displaystyle M=\left\{q: q<\sqrt{3},\ q\in\mathbb Q\right\}$$

• #### Variant 5

$$M=\{\sin x: x\in [ 0, 2\pi)\}$$

• #### Variant 6

$$M=\{\sin x: x\in (0, 2\pi)\}$$

• #### Variant 7

$$M=\{\sin x: x\in (0, \pi)\}$$

• #### Variant 8

$$\displaystyle M=\left\{1-\frac1{n^2},n\in\mathbb N\right\}$$

• #### Variant 9

$$\displaystyle M=\left\{\frac{n-1}n,n\in\mathbb Z\setminus\{0\}\right\}$$

• #### Variant 10

$$\displaystyle M=\left\{\frac{p}{p+q},p,q\in\mathbb N\right\}$$

• #### Variant 11

$$\displaystyle M=\left\{\frac{n+(-1)^n}n,n\in\mathbb N\right\}$$

• #### Variant 12

$$\displaystyle M=\left\{n^{(-1)^n},n\in\mathbb N\right\}$$

• #### Variant 13

$$M=\{n^2-m^2: n,m\in \mathbb N\}$$

• #### Variant 14

$$M=\{n^2-m^2: n,m\in \mathbb N, n>m\}$$

• #### Variant 15

$$M=\{n^2-m^2: n,m\in \mathbb N, n\le m\}$$

• #### Variant 16

$$M=\{2^{-n}+3^{-n}: n\in \mathbb N\}$$

• #### Variant 17

$$M=\{2^{-n}+3^{-n}: n\in \mathbb Z\}$$

• #### Variant 18

$$M=\{5^{(-1)^j3^k}: j,k\in \mathbb Z\}$$

• #### Variant 19

$$\displaystyle M=\left\{\cos\left(\frac{n+1}n\pi\right),n\in\mathbb N\right\}$$

• #### Variant 20

$$\displaystyle M=\left\{\cos\left(\frac{n+1}n\pi\right),n\in\mathbb N, n \text{ sudé}\right\}$$

• #### Variant 21

$$\displaystyle M=\left\{\cos\left(\frac{n+1}n\pi\right),n\in\mathbb N, n \text{ liché}\right\}$$