Real sequences/Convergence and boundedness/Section

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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.