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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.

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