1. ## sequence question

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?

2. Have you studied the concept of Cauchy Sequences?
If you have, then you know that if $\displaystyle (a_n)$ converges then $\displaystyle (a_n)$ is a Cauchy sequence.

Being a Cauchy sequence what does that tell you about the difference $\displaystyle \left| {a_n - a_{n + 1} } \right|$?
In other words, does it mean that $\displaystyle {\left( {a_n - a_{n + 1} } \right)}$ converges?

3. Thanks for answering, but no we haven't studied those yet.

Are you saying it does converge?

4. Originally Posted by CindyMichelle
Thanks for answering, but no we haven't studied those yet.
Well then here is another way to think about it.
If $\displaystyle (a_n)$ converges then almost all of its terms get very close to the limit value. Does that make sense? Therefore almost all of its terms have a difference close to zero. So $\displaystyle {\left( {a_n - a_{n + 1} } \right)}$ converges to zero.

5. How about an = (-1)^(n). Then an+1 = (-1)^(n+1).
limit as n approaches inf of abs(an - an+1) = 2.
limit as n approaches (an - an+1) does not exist.

6. Does this not contradict what you are saying?