# Upper and lower limit proof (liminf/sup)

Jesssa
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?

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voko
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.

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Jesssa
The conditions are that they are both bounded real sequences,

Hmm

voko
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.

Jesssa
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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voko
This looks good to me.