1.：Strict definition of the limit of a sequence

　　Definition: Given a sequence and a constant , if for any given positive number ( no matter how small it is) , there is always a positive integer , such that for all when , the inequality

If all are true, then is said to be the limit of the sequence  , or the sequence is said to converge to  , which is recorded as

or

　　If a sequence has no limit, it is said to be divergent.

　　Note 1: " For any given positive number " in the definition actually expresses the meaning that approaches infinitely in the definition is related to any given positive number .

　　Note 2: By definition, the limit of a sequence has nothing to do with the value of the previous finite term of the sequence.

　　2. Geometric interpretation of sequence limits

　　Geometric interpretation of : express the constant and the sequence on the number axis, and make the field on the number axis ( Figure 1 ) .

Figure 1：  [ Animation ]

Note that the inequality is equivalent to

,

Therefore, the limit of the sequence  is  , which means geometrically: when  , all points  fall within the open interval  . There are at most  points falling outside this interval .

　　Note 3: The definition of the limit of a sequence does not provide a method for finding the limit. It only provides a method for demonstrating that the limit of a sequence is . It is often called argument method. The argument steps are:

　　(1) For any given positive number ;

        (2) Starting from the inequality  , analyze and work backwards to deduce  . The purpose is to specifically find an  that satisfies the definition . In this way, during the process of analyzing and working backwards, we can use common inequalities and their properties to simplify and deduce  by scaling the inequality .

　　(3) At this time, we only need to take to ensure that when , there is always

.

Finally, state the conclusion in language to complete the proof.