Find a sequence $a_n$ such that $\lim_{n \to \infty}|a_{n+1} - a_n | = 0$ while the sequence does not converge Find a sequence $a_n$ such that $\lim_{n \to \infty}|a_{n+1} - a_n | = 0$ while the sequence does not converge.

I am thrown for a loop. For a sequence not to converge it means that for all $\epsilon \geq 0$ and for all $N$, when $n \geq N$, $|a_n - L| \geq \epsilon$ as $n \longrightarrow \infty$

or I could change some terms and state it in terms of a Cauchy sequence, but either way how can the terms have a distance of 0 but the sequence not be consider converging?  hints??
Take $a_n=\log(n)$; then

$$
|a_{n+1}-a_n|=\log(n+1)-\log(n) = \log(1+\frac{1}{n})\to 0,
$$

but the sequence diverges. In the definition of Cauchy sequences it is asserted that $|a_m-a_n|\to 0$, but you are free to choose any such $m,n$ independent of each other. In your case $m=n+1$.
How about $a_n = \sum_1^n \frac{1}{m}$? The sequential differences are of course equal to $\frac{1}{n}$, but the limit of the sequence is the sum of all such fractions. 

Of course, that's just one concrete example of what @Jake said.