I cant see the logic of thes definitions

[transgalactic]

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

 

[matt grime]

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.

 

[transgalactic]

you can enlarge it by clicking on it.
and yes its is the supremum of the limits
here is the definition
http://img23.imageshack.us/img23/6383/18151267.th.gif

actually it both thing
so my definition is correct

 

[matt grime]

you can enlarge it by clicking on it.

No I can't.

and yes its is the supremum of the limits

That makes no sense, and is in fact wrong: just google the definition.

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

is one that is correct (even though it is wiki).

here is the definition
http://img23.imageshack.us/img23/6383/18151267.th.gif

That is again, a small little thumbnail that cannot be usefully zoomed in on, nor clicked on in my browser.

 

[transgalactic]

her is the definition that i got
http://img23.imageshack.us/img23/6383/18151267.gif

 

[Focus]

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.

 

[matt grime]

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...