The sign-preservation theorem of convergent sequences and its corollary

　　Theorem 3: If , and  (or ) , then there exists a positive integer such that when , there is always

( or ) .

　　Proof : First prove the situation of  . According to the definition of sequence limit , for  , there is a positive integer  When  , there is

 ,

Using the basic absolute value inequality , we get

,

so

.

        Similarly, the situation of  can be proved . Certification completed.

　　Corollary 2：If the sequence has ( or ) starting from a certain item , and

,

but

( or ) .

　　Proof : Prove that the sequence  has  starting from the  -th item .

　　Use proof by contradiction : if , then according to the above theorem , there is a positive integer , and when , there is . Pick

,

When , there is , but by assumption there is , which is a contradiction. Therefore there must be

.

　　In the same way, it can be proved that the sequence starts from a certain item with . Certification completed.