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Bounded set
From Citizendium, the Citizens' Compendium
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 subsequence that 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.
