Math and science::Analysis::Tao::09. Continuous functions on R

# Bounded sets (of reals)

A subset \( X \) of the real line is said to be*bounded*if we have \( X \subseteq [-M, M] \) for some real number \( M > 0 \).

Not to be confused with bounded sequences.

### Example

The interval \( [a, b] \) is bounded for any real numbers \( a \) and \( b \), as it is contained inside \( [-M, M] \) where \( M := max(|a|, |b|) \).

The half-infinite intervals and the doubly infinite interval are not bounded.

The sets \( \mathbb{N}, \mathbb{Z}, \mathbb{Q} \text{ and } \mathbb{R} \) are all unbounded.