Linked Questions

10
votes
4answers
4k views

Terms that get closer and closer together [duplicate]

Possible Duplicate: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy? I was thinking about sequences where it appears the terms get closer and closer ...
7
votes
3answers
4k views

Weaker version of Cauchy sequence criteria [duplicate]

Possible Duplicate: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy? my question is this: The following definition is weaker than the definition of ...
4
votes
2answers
143 views

Does every sequence that satifies this requirement converge? [duplicate]

Possible Duplicate: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy? If we have a sequence $(x_n)_n$ that satisfies this rule: $\forall \varepsilon &...
3
votes
5answers
140 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&...
3
votes
3answers
159 views

$\{x_n\}$ be a bounded sequence of distinct real numbers , $|x_{n+1}-x_n|<|x_n-x_{n-1}|,\forall n\in \mathbb N$ , then is $\{x_n\}$ convergent? [duplicate]

Let $\{x_n\}$ be a bounded sequence of distinct real numbers such that $|x_{n+1}-x_n|<|x_n-x_{n-1}|,\forall n\in \mathbb N$ , then is it true that $\{x_n\}$ converges ? The motivation for this ...
6
votes
4answers
3k views

Pseudo-Cauchy sequence

I have never seen this terminology before, so I will provide the given definition. A Pseudo-Cauchy sequence is : A sequence $(a_n)$ if for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such ...
5
votes
3answers
3k views

Example of a divergent sequence

I want to produce a divergent sequence for which $|x_n - x_{n-1}| \to 0$. So far, I've only been able to show that $$\frac{x_n}{n} \to 0$$, which doesn't really help.
5
votes
3answers
151 views

$\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$

I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure. For a sequence $\{a_n\}$, there exists a limit $a$ such that $\lim_{n\to\infty} a_n=a$ if and only if for ...
1
vote
2answers
194 views

Show that sequences satisfying these two conditions need not be and must be Cauchy, respectively

Given a sequence $(a_n)$ prove that a) If it is known that $|a_{n+1}-a_n|< 1/n$ for all $n$, show that $(a_n)$ need not be a Cauchy Sequence. b) If it is known that $|a_{n+1}-a_n| < 1/2^n$ ...
3
votes
0answers
1k views

cauchy sequence and necessary and sufficient condition for convergence

Question: Show that for a sequence $\{x_m\}$ of real numbers to be a Cauchy sequence, it is necessary, but not sufficient that $|x_{m+1}-x_m|$ converges to zero. This is how I proved that it is ...
0
votes
3answers
142 views

Is there a divergent sequence such that $(x_{k+1}-x_k)\rightarrow 0$?

Is there a divergent sequence such that $\lim_{k\rightarrow\infty}(x_{k+1}-x_k)=0$?
4
votes
2answers
101 views

Proving convengent sequence theorem.

When $n$ approach to infinity prove that if $$ \lim(a_{n+1}-a_n))= 0,$$ then $a_n$ is convergent. I can prove the converse of this theorem is true but I can't prove this one. I know that since $$ ...
1
vote
1answer
197 views

Bounded sequence implies convergence

Let $(a_n)_{n\in\mathbb{N}}, (e_n)_{n\in\mathbb{N}}, (p_n)_{n\in\mathbb{N}}, (r_n)_{n\in\mathbb{N}}$ be nonnegative sequences in $\mathbb{R}$ with $(a_n)_{n\in\mathbb{N}}\in\ell^1$ and $(e_n)_{n\in\...
1
vote
1answer
76 views

Is there an example of ward compact set?

A set $A$ is ward compact if every sequence $(a_n) $ in $A$ has a quasi-Cauchy subsequence. A sequence $(x_n)$ is said to be quasi Cauchy if $x_n-x_{n+1}\rightarrow 0$. We know that $a_n=\sqrt n$ is a ...
0
votes
0answers
62 views

Given a sequence in R is it sufficient to prove limit d(xn,xn+1) is 0 to show that it is cauchy?

I saw a proof in a fixed point theorem which showed that d(xn,xn+p) is 0 as n goes to infinity hence the sequence is cauchy. using triangle inequality if d(xn,xn+p) less than equal to d(xn,xn+1)+d(...