Bounded set

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In mathematics, a bounded set is any subset of a normed space whose elements all have norms which are bounded from above by a fixed positive real constant. In other words, all its elements are uniformly bounded in magnitude.

Formal definition

Let X be a normed space with the norm \|\cdot\|. Then a set A \subset X is bounded if there exists a real number M > 0 such that \|x\|\leq M for all x \in A.

Theorems about bounded sets

Every bounded set of real numbers has a supremum and an infimum. It follows that a monotonic sequence of real numbers that is bounded has a limit. A bounded sequence that is not monotonic does not necessarily have a limit, but it has a monotonic subsequence, and this does have a limit (this is the Bolzano–Weierstrass theorem).

The Heine–Borel theorem states that a subset of the Euclidean space Rn is compact if and only if it is closed and bounded.

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