How to Manipulate Lim Sups and Lim Infs in Proofs?

[cmurphy]

Hello,

I have several questions regarding lim sups and lim infs. I have a couple of proofs that I need to do, and I'm not sure where to start, because I don't have a good understanding of how to "play" with the definition; lim sup sn = lim N -> infinity sup{sn: n > N}.

Any suggestions?

An example of a problem I'm struggling with is:

Show that lim sup(sn + tn) is less than or equal to lim sup sn + lim sup tn for bounded sequences (sn) and (tn).

Similarly, how could I show that lim sup(sntn) is less than or equal to (lim sup sn)(lim sup tn), where (sn) and (tn) are bounded sequences of nonnegative integers?

Thanks in advance,
Colleen

 

[mathman]

For any e, there is an N so that sn<=e+limsupsn and tn<=e+limsuptn, for all n>=N. Therefore sn+tn<=limsupsn+limsuptn+2e for all n>=N. I'm sure you can finish it. The same trick can be used for the product limsup.

 

[cmurphy]

I understand what you've done, but I don't actually see how to proceed. Since
sn + tn - 2e <= lim sup sn + lim sup tn, how do you get lim sup (sn + tn)? I think this might go back to the problem that I'm having with understanding how to manipulate lim sup. Conceptually, I understand what it means, but in terms of manipulating this for a proof, I get lost.