Linked Questions

3
votes
5answers
162 views

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&...
19
votes
5answers
2k views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy? I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\...
4
votes
3answers
2k views

A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence
2
votes
1answer
1k views

An example of a bounded pseudo Cauchy sequence that diverges? [duplicate]

Harmonic series diverges and pseudo Cauchy however it's not bounded. So how can I find such a sequence? A sequence $(s_n)$ is pseudo-Cauchy if, for all $\xi>0$, there exists an $N$ such that if $n ...
3
votes
2answers
945 views

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. ...
1
vote
2answers
437 views

If a set is Pseudo-Cauchy, then it is NOT necessarily Cauchy [duplicate]

The following is the definition of a pseudo-Cauchy sequence: A sequence $a_n$ is pseudo Cauchy if $\forall$ $\epsilon > 0$, $\exists \ N \in \mathbb N$ such that whenever $n \ge N$, $|a_{n+1} - ...
0
votes
3answers
119 views

A sequence $\{a_n\}$ that diverges but $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$ [duplicate]

What is a sequence $\{a_n\}$ that diverges and $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$
2
votes
3answers
153 views

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy [duplicate]

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy ? I tried to find such sequence $x(n)=1/2,1/3,1/2,1/3,1/4,1/2,1/3,1/4,1/5,,,,$ it's not Cauchy since it is ...
2
votes
2answers
87 views

Finding a sequence a with $\lim_{ n\to ∞} (a_{n+1}-a_n)=0$ a:divergent [duplicate]

Question is in the title. I would appreciate any help with this as I am a bit clueless.
2
votes
2answers
295 views

Cauchy sequences and condition $X_{n+1} – X_n\to 0$

Let $\{X_n\}$ be a sequence and suppose that the sequence $\{X_{n+1} – X_n\}$ converges to $0$. Give an example to show that the sequence $\{X_n\}$ may not converge. Hence, the condition that $|X_n-...
-1
votes
1answer
72 views

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 ...