Real sequences/Convergence and boundedness/Section
A subset of the real numbers is called bounded if there exist real numbers such that.
In this situation, is also called an upper bound of and is called a lower bound of . These concepts are also used for sequences, namely for the image set, the set of all members . For the sequence , , is an upper bound and is a lower bound.
A convergent real sequence is bounded.
Let be the convergent sequence with as its limit. Choose some . Due to convergence there exists some such that
So in particular
Below there are ony finitely many members, hence the maximum
is welldefined. Therefore is an upper bound and is a lower bound for .
It is easy to give a bounded but not convergent sequence.
The alternating sequence
is bounded, but not convergent. The boundedness follows directly from for all . However, there is no convergence. For if were the limit, then for positive and every odd the relation
holds, so these members are outside of this -neighbourhood. In the same way we can argue against some negative limit.