Prove that the sequence is divergent.
Prove：By contradiction : Suppose the given sequence converges to , that is
,
then it is defined by the limit . For , there exists such that when , there is always
,
using the basic absolute value inequality , that is, when , there is
,
the intervalThe length of is . And takes two numbers endlessly , and it is impossible to be at the same time, which is a contradiction. Therefore, the sequence is divergent .
Note : This example also shows that bounded sequences do not necessarily converge.
