Have you studied the concept of Cauchy Sequences?
If you have, then you know that if converges then is a Cauchy sequence.
Being a Cauchy sequence what does that tell you about the difference ?
In other words, does it mean that converges?
Let (an) be a sequence such that lim as n approaches infinity of (an + an+1) exists and lim as n approaches infinity of (anan+1) exists.
I'm pretty sure that lim as n approaches infinity of (an - an+1) does not exist (the textbook seems to imply this), yet I can't think of a counterexample. Can someone give me a sequence where this limit does not exist?