lim sups and lim infs

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.