Bounded set

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.