# 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 . Then a set is bounded if there exists a real number *M* > 0 such that for all .

## 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 **R**^{n} is compact if and only if it is closed and bounded.