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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. 


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.