Q:

If $|x_{n+1} -x_n|<AC^n$ then $(x_n)$ is Cauchy

Let ${x_n}$ be a sequence such that there exist $A>0$ and $C\in (0,1)$ for which $|x_{n+1} -x_n|<AC^n$, for any $n\geq 1$. Show that $\{x_n\} $ is Cauchy. Is this conclusion still valid if we assume only $\lim_{n\to\infty} |x_{n+1}-x_n|=0? $

A:

Note, that for $m > n$, we have 
\begin{align*}
  \def\abs#1{\left|#1\right|}\abs{x_m - x_n}&= \abs{\sum_{k=n}^{m-1} x_{k+1} - x_k}\\
  &\le \sum_{k=n}^{m-1}\abs{x_{k+1} - x_k}\\
 &\le A \sum_{k=n}^{m-1} C^k\\
  &= AC^n \sum_{k=0}^{m-n-1} C^k\\
  &= AC^n \frac{C^{m-n} - 1}{C - 1}\\
  &\le \frac{AC^n}{1-C}\\
  &\to 0, n \to \infty.
\end{align*}
For your additional question, have a look at $x_n = \sum_{k=1}^n \frac 1k$. 

