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Upper and lower limit proof (liminf/sup)

  • Thread starter Jesssa
  • Start date
  • #1
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0
Hey,
So I have been working on this question for quite a while now and I'm at this point.
I just wanted to check if everything was okay, I never feel confident with questions like this. Here is the question and my working,

http://img404.imageshack.us/img404/8436/afafafa.jpg [Broken]

Is there anything wrong?

Thanks in advanced.
 
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Answers and Replies

  • #2
6,054
390
Are there are conditions on Xn and Yn? The inequality is not true in general. If we set Xn - Yn = a = const, then Xn + Yn = 2, so lim inf {Xn + Yn} = lim sup {Xn + Yn} = a, but lim inf Xn and lim sup Xy may not exist.
 
Last edited:
  • #3
51
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The conditions are that they are both bounded real sequences,

Hmm
 
  • #4
6,054
390
OK, bounded makes it correct.

You have the first part correctly.

The second, however, is wrong. You show that Ix + Sy <= Sx + Sy. That's true, but that does not mean that Ix + Sx <= Sw.
 
  • #5
51
0
I have found another way, do you think this works?

http://img211.imageshack.us/img211/1769/89852645.jpg [Broken]

the last part of the first line sup(inf(x)+y) = inf(x) + sup(y), because inf(x) will just be some number,
 
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  • #6
6,054
390
This looks good to me.
 

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