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

Is there anything wrong?

Thanks in advanced.

 

[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.

 

[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

the last part of the first line sup(inf(x)+y) = inf(x) + sup(y), because inf(x) will just be some number,

 

[voko]

This looks good to me.