## Suprema and infima of concrete sets

### Task number: 2824

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

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