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### 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 ...
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### 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 ...
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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: $\forall \varepsilon &... 5answers 140 views ### Does$|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. $$\forall \epsilon&... 3answers 159 views ### \{x_n\} be a bounded sequence of distinct real numbers , |x_{n+1}-x_n|<|x_n-x_{n-1}|,\forall n\in \mathbb N , then is \{x_n\} convergent? [duplicate] Let \{x_n\} be a bounded sequence of distinct real numbers such that |x_{n+1}-x_n|<|x_n-x_{n-1}|,\forall n\in \mathbb N , then is it true that \{x_n\} converges ? The motivation for this ... 4answers 3k views ### Pseudo-Cauchy sequence I have never seen this terminology before, so I will provide the given definition. A Pseudo-Cauchy sequence is : A sequence (a_n) if for any \epsilon > 0 there exists N \in \mathbb{N} such ... 3answers 3k views ### Example of a divergent sequence I want to produce a divergent sequence for which |x_n - x_{n-1}| \to 0. So far, I've only been able to show that$$\frac{x_n}{n} \to 0$$, which doesn't really help. 3answers 151 views ### \lim_{n\to\infty} a_n=a if and only if \forall p\in \Bbb N, \lim_{n\to\infty} |a_{n+p}-a_n|=0 I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure. For a sequence \{a_n\}, there exists a limit a such that \lim_{n\to\infty} a_n=a if and only if for ... 2answers 194 views ### Show that sequences satisfying these two conditions need not be and must be Cauchy, respectively Given a sequence (a_n) prove that a) If it is known that |a_{n+1}-a_n|< 1/n for all n, show that (a_n) need not be a Cauchy Sequence. b) If it is known that |a_{n+1}-a_n| < 1/2^n ... 0answers 1k views ### cauchy sequence and necessary and sufficient condition for convergence Question: Show that for a sequence \{x_m\} of real numbers to be a Cauchy sequence, it is necessary, but not sufficient that |x_{m+1}-x_m| converges to zero. This is how I proved that it is ... 3answers 142 views ### Is there a divergent sequence such that (x_{k+1}-x_k)\rightarrow 0? Is there a divergent sequence such that \lim_{k\rightarrow\infty}(x_{k+1}-x_k)=0? 2answers 101 views ### Proving convengent sequence theorem. When n approach to infinity prove that if$$ \lim(a_{n+1}-a_n))= 0,$$then a_n is convergent. I can prove the converse of this theorem is true but I can't prove this one. I know that since$$ ... 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\...
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 ...