I cant see the logic of thes definitions

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Homework Help Overview

The discussion revolves around the concept of limit superior (lim sup) in the context of sequences in mathematical analysis. Participants are exploring the definitions and implications of lim sup, questioning the logic behind its interpretation and application.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the definition of lim sup, with some asserting it is the supremum of the limits while others argue it is the limit of the supremums. There are questions regarding the validity of definitions presented through images, and whether the definitions provided are accurate or helpful.

Discussion Status

The discussion is active, with participants presenting differing interpretations of the lim sup concept. Some participants have provided external references, such as Wikipedia, to support their claims, while others express confusion and seek further clarification. There is no explicit consensus on the definitions being debated.

Contextual Notes

Participants note issues with the clarity of the definitions presented through images, which are described as too small to read. The conversation reflects a mix of personal interpretations and external sources, indicating a lack of agreement on the foundational definitions of lim sup.

transgalactic
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http://img21.imageshack.us/img21/3070/11650282.th.gif

on the limsup case

each time we take out the biggest member of the sequence .
so it goes to the smallest memeber
how its supposed to be the supremum of the limits
 
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It isn't the supremum of the limits.

Lim sup means the limit of the supremums. I.e. the reverse of what you were thinking.

You .gif is too small to be read, at least by me.
 
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her is the definition that i got
http://img23.imageshack.us/img23/6383/18151267.gif
 
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transgalactic said:
her is the definition that i got
http://img23.imageshack.us/img23/6383/18151267.gif

He is using an example, this is not a definition. limsup is the limit supremum, that is the limit of the supremums (supremums are a decreasing function on n). In that case the sequence oscillates, so the supremum is 1 and infemum is -1, hence limsup=1 and liminf=-1.

The wiki article is correct. I think the picture is more helpful to remember these.
 
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It doesn't even make sense to take the supremum of limits. Why?

If x_n converges to x, then so does every subsequence, so sups of any sequence of limits can only be x. Conversely, if x_n does not converge, then you don't have limits of which you can take the sups...
 

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