(Subseqeunces) Limit Superior and Limit Inferior

 

1. divergence, the improper accumulation value

 

2. limsup, liminf definition

 

 

 

 

3. limsup, liminf lemma

- Because the sequence is monotonically decreasing and bounded, we also know that the limit is going to be the same thing as the infimum of sup s_k

 

No matter how big number k is, the largest accumulation value is not changed.

 

Another definition of limsup and liminf

위랑 같은 식

 

4. limsup, liminf lemma 2

 

극한의 정의와 유사하나 절대값이 아니고 infimum이 존재한다.

- 11/22/17 The third version of the limsup definition underscores its relationship with the limit definition: replacing the requirement that a tail of the sequence remains within an epsilon strip (abs(s_n - L) < epsilon) with the weaker condition that it does not crash through the ceiling of that strip.

 

 

5. Examples for limsup and liminf