Does $|a_n-a_{n+1}|\to 0$ imply $(a_n)$ is Cauchy? My textbook has this problem as a kind of "concept check", where one is supposed to find a counterexample to the following statement: 
A sequence of real numbers is cauchy iff. 
$$ \forall \epsilon>0, \, \exists N \in \mathbb{N}, \, \forall n \geq N: |a_n-a_{n+1}|< \epsilon $$
I know that a sequence of real numbers is cauchy if 
$$ \forall \epsilon>0, \, \exists N \in \mathbb{N}, \, \forall n,m \geq N: |a_n-a_m|< \epsilon $$ 
Finding such a counterexample is probably trivial, but I haven't been able to think of one. 

My go-to counterexample here is to take partial sums of the harmonic series. That is, $$a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$$
Obviously the difference between $a_{n-1}$ and $a_n$ is just $\frac{1}{n}$, which goes to $0$, but this sequence does not converge. 
