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