Quasi-Cauchy sequences

Question

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:

1. Quasi-Cauchy sequence of real numbers is Cauchy if and only if it has exactly one cluster point.
2. 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 :)

Answer 1

1. Any sequence of real numbers which has exactly one cluster point is a Cauchy.
2. If $(a_n ) $ is not Cauchy sequence then it is not convergent. Hence there exists a subseqences $(x_{n_k } )$, $(y_{m_k} )$ of $(a_n )$ such that $x_{n_k }\to x, y_{n_k }\to y$ and $x\neq y $ and $n_k \neq m_k $ for all $k.$ Consider a subsequence $(z_k)$ of $(a_n)$ defined by $z_{2k-1} =x_{n_k} , z_{2k} =y_{m_k}.$ Then $$\lim_{k\to\infty} |z_k -z_{k+1}|=\lim_{k\to\infty} |x_{n_k} -y_{m_k}|=|x-y|\neq 0$$ hence the subsequence $(z_k)$ is not quasi-Cauchy.

Answer 2

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 $q$.