limsup-liminf

limsup-liminf - Limit sup and limit inf. Introduction In...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Limit sup and limit inf. Introduction In order to make us understand the information more on approaches of a given real sequence a n n 1 , we give two definitions, thier names are upper limit and lower limit. It is fundamental but important tools in analysis. Definition of limit sup and limit inf Definition Given a real sequence a n n 1 ,wede f ine b n sup a m : m n and c n inf a m : m n . Example 1 1 n n 1 0,2,0,2,. .. ,sowehave b n 2and c n 0 for all n . Example  1 n n n 1 1,2, 3,4,. .. ,sowehave b n  and c n  for all n . Example n n 1 1, 2, 3,. .. ,sowehave b n n and c n  for all n . Proposition Given a real sequence a n n 1 , and thus define b n and c n as the same as before. 1 b n  ,and c n  n N . 2 If there is a positive integer p such that b p  , then b n   n N . If there is a positive integer q such that c q  , then c n   n N . 3 b n is decreasing and c n is increasing. By property 3, we can give definitions on the upper limit and the lower limit of a given sequence as follows. Definition Given a real sequence a n and let b n and c n as the same as before. (1) If every b n R , then inf b n : n N is called the upper limit of a n , denoted by lim n sup a n . That is, lim n sup a n inf n b n
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.

Page1 / 4

limsup-liminf - Limit sup and limit inf. Introduction In...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online