TITLE: Quasi-Cauchy sequences
QUESTION [8 upvotes]: Sequence $(x_n)$ is called quasi-Cauchy if $\lim_{n\rightarrow\infty}|x_{n+1}-x_n|=0.$
I need help proving the following theorems:

Quasi-Cauchy sequence of real numbers is Cauchy if and only if it has exactly one cluster point.
Sequence of real numbers is Cauchy if and only if every subsequence is quasi-Cauchy.

I understand the implications to the right (they are trivial), but have trouble proving the opposite way. Any help would be appreciated :)

REPLY [6 votes]: For the R-to-L of 1.  suppose $(x_n)_n$ is quasi-Cauchy and has a cluster point $p$  but $(x_n)_n$ is not Cauchy. Then $(x_n)_n$ does not converge to $p,$ so there exists $r>0$ such that the set $$S=\{n: |x_n-p|\geq r\}$$ is infinite.
Take $n_0$ such that $n> n_0\implies |x_n-x_{n+1}|<r/3.$ Now the set $$T=\{n:|x_n-p|< r/3\}$$ is also infinite because $p$ is a cluster point of $(x_n)_n$.
For $n_0<n\in T$ there exists $n'> n$ with $n'\in S.$ So for $n_0<n\in T$ let $f(n)$ be the least $n'>n$ such that $n'\in S.$  
Observe that the set $U=\{f(n): n_0<n\in T\}$ is infinite .   We have  $$m\in U\implies (\;|x_m-p|\geq r \;\land \;|x_m-x_{m-1}|<r/3 \;\land\; |x_{m-1}-p|<r\;).$$  So for $m\in U$ we have $$x_m\in V=[p-4r/3,p-2r/3]\;\cup \; [p+2r/3, p+4r/3].$$ But there are infinitely many $m\in U,$ so $(x_n)_n$ must have a cluster point $q\in V$, and obviously  $q\ne p.$ 
Summary: A quasi-Cauchy sequence $(x_n)_n$ with exactly one cluster point $p$ must be convergent to $p$, otherwise $(x_n)_n$ would have another cluster point $q$.