11 questions linked to/from Pseudo-Cauchy sequence
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### Does $|a_n-a_{n+1}|\to 0$ imply $(a_n)$ is Cauchy? [duplicate]

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&...
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### Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
### If $|x_{n+1} -x_n|<AC^n$ then $(x_n)$ is Cauchy [closed]
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 ...