suppose
then
certainly doesn't converge. It is however bounded. I don't think you can play the same trick if {a} is unbounded. I'd have to think about it.
Let . According to the question, . As romsek showed, the convergence of has nothing to do with the convergence of . But, we can use to look at properties of .
Check out . By limit laws, the limit of the difference is the difference of the limits, so long as both limits exist. Since both do, we know that . You can do the same for . So, if you define sequences , perhaps you can show that both of these sequences are Cauchy (using the previous results), and hence they both converge. Since convergent sequences are bounded, the original sequence must be bounded, as well.
Hmmmm, maybe we can take the sequence a_{n}=n , so we can say that a_{n+1-}a_{n} converges to 1? but a_{n} isn't bounded? Is this possible? i mean i know infinite minus infinite in limits is a problem but the distance between them should be 1 and it shouldn't be a problem right?(I got confused , this is completely wrong