I answered a question but I don't know if my answer is correct so I would appreciate if someone can confirm it. Thank you.
By the way I did some mistake when I wrote it. its a_{n+1}-a_{n}
suppose
$\displaystyle a_n=(-1)^n$
then
$\displaystyle \forall n (a_{n+1}+a_n)=0$
certainly $\displaystyle a_n$ 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.
Looks like I'm wrong.
$\displaystyle a_n=(-1)^n log(n)$
clearly $\displaystyle a_n$ is unbounded.
$\displaystyle a_n+a_{n+1}=(-1)^n log(n) + (-1)^{n+1} log(n+1)$
let n be even
$\displaystyle a_n+a_{n+1}=log(n)-log(n+1)=log(\frac{n}{n+1})\rightarrow log(1)=0$
similar if n is odd
So you can manage to make the sum of adjacent elements converge to 0 even while the sequence itself is unbounded.
Let $\displaystyle b_n = a_{n+1} + a_n$. According to the question, $\displaystyle \lim_{n \to \infty} b_n = 0$. As romsek showed, the convergence of $\displaystyle b_n$ has nothing to do with the convergence of $\displaystyle a_n$. But, we can use $\displaystyle b_n$ to look at properties of $\displaystyle a_n$.
Check out $\displaystyle \lim_{n \to \infty} (b_{n+1} - b_n)$. 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 $\displaystyle \lim_{n \to \infty} (b_{n+1} - b_n) = \lim_{n \to \infty} (a_{n+2} - a_n) = 0$. You can do the same for $\displaystyle \lim_{n \to \infty} (b_{n+2} - b_{n+1}) = \lim_{n \to \infty} (a_{n+3} - a_{n+1}) = 0$. So, if you define sequences $\displaystyle c_n = a_{2n}, d_n = a_{2n+1}$, 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