# Bounded Subset of Real Numbers/Examples

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## Examples of Bounded Subsets of Real Numbers

### Example 1

The subset $S$ of the real numbers $\R$ defined as:

- $S = \set {1, 2, 3}$

is bounded both above and below.

Some upper bounds of $S$ are:

- $100, 10, 4, 3$

Some lower bounds of $S$ are:

- $-27, 0, 1$

### Example 2

The subset $T$ of the real numbers $\R$ defined as:

- $T = \set {x \in \R: 1 \le x \le 2}$

is bounded both above and below.

Some upper bounds of $T$ are:

- $100, 10, 4, 2$

Some lower bounds of $T$ are:

- $-27, 0, 1$

### Reciprocals of Positive Integers

The subset $T$ of the real numbers $\R$ defined as:

- $T = \set {\dfrac 1 n: n \in \Z_{>0} }$

is bounded both above and below.

We have that:

- $\sup T = 1$
- $\inf T = 0$

where $\sup T$ and $\inf T$ denote the supremum and infimum of $T$ respectively.

We also have:

- $\sup T \in T$
- $\inf T \notin T$