# Real sequences/Convergence and boundedness/Section

## Definition

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.

## Lemma

A convergent real sequence is bounded.

### Proof

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.

## Example

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.