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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

A sequence $\{x_n\}$ in an arbitrary metric space, and in particular the space $\Bbb{R}$, is called Cauchy if the terms of the sequence become arbitrarily close together; that is, for every $\epsilon > 0$, there exists an $N$ such that

$$n, m \ge N \implies d(x_n, x_m) < \epsilon$$

where $d$ is the distance function for the metric space. In the particular case of the real numbers, this condition becomes

$$n, m \ge N \implies |x_n - x_m| < \epsilon$$

A complete metric space is a metric space in which every Cauchy sequence is convergent; this gives an alternate definition of convergence of a sequence that does not rely on the limiting value.

Source: the Cauchy sequence article on Wikipedia.

2431 questions
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Existence of Cauchy and convergent sequences

I understand the definition of what is a Cauchy sequence, and what is a convergent sequence,so my question is in many topics involving functional analysis and/or Sobolev spaces, it is assumed there exists a Cauchy sequence, or a convergent sequence,…
sandy kuks
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Merten's theorem on cauchy products

Suppose we know that $$\{a_n\}$$ is a sequence such that $$\sum_{n=0}^{\infty}a_n=0$$ and that for some $N$ quite large, we have that $a_n=0$ for any $n\geq N$. Then we notice that the infinite series of $a_n$ converve absolutely. If we were to…
user81654
  • 91
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How to prove that the following sequence converges?

Let $u_1$=2 and $u_{n+1}=2+\frac{1}{u_n}$ for $n\geq 1$. Prove that the sequence converges to $\sqrt{2} +1$. I have no clue how to do it,probably using Cauchy sequence definition. The hint given in textbook is:…
Natasha J
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How should I prove that the following sequence is cauchy?

I have to prove from definition that the following sequence is Cauchy: ${1+\frac{1}{1!}+\frac{1}{2!}...+\frac{1}{n!}}$. Definition of Cauchy sequence: A sequence $(a_n)$ is said to be a Cauchy sequence if given $\epsilon>0$, however small, there…
Natasha J
  • 825
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Is it a Cauchy sequence?

Let $(X,d)$ be a metric space and $(x_n)$ is a sequence in $X$. Then $\sup_{p\ge 1} d(x_n,x_{n+p}) \rightarrow 0$ as $n\rightarrow \infty$ implies $(x_n)$ is a Cauchy sequence. Any hints or counterexample to prove or disprove this implication?…
Neon
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Induction on a contractive sequence

By definition I know that a contractive sequence is called contracting if there exists $l\in (0,1)$ such that for $n>N$: $$|x_{n+2}-x_{n+1}|\le l |x_{n+1}-x_n|;\forall n \in\mathbb{N}$$ There is an inequality $|x_{n+1}-x_n| \le l^{n-1} |x_2-x_1|$…
Aurora Borealis
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If $a_k$ is a sequence that converges to $1/2$, prove that $b_k = 1/(1-a_k)$ is Cauchy

The question asks to prove directly from the definition of a Cauchy sequence that $b_k$ is Cauchy, but I am hopelessly confused, these are evidently series approaching infinity
user825131
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A construction in a non Cauchy Sequence

Let $(y_n)$ be a sequence which is not Cauchy. Let $\lim_{n\to\infty} d(y_n,y_{n+1})=0$ and $d(y_n,y_{n+1})$ is a decreasing sequence. Then there exists an $\epsilon>0$ such that for every $n\in \mathbb N$, there exists an odd integer $q(n)\in…
Earth
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Cauchy sequences in $\mathbb{C}$ does not imply that the function converges.

I am currently working on a proof and I have the current properties: A Cauchy sequence $\{ f_n\}_{n\in \mathbb{N}} $ of functions in some space $X$ with the property that $|(f_n - f_m)(x)|<\epsilon$ for any given $\epsilon > 0$ which I have used to…
Hugh Entwistle
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Prove that the sequence $x_n = 1 +\frac{ sin (n+ \pi) }{n} $ is a cauchy sequence

Prove that the sequence $x_n = 1 +\frac{ sin (n+ \pi) }{n} $ is a cauchy sequence using the definition: $$\forall \epsilon>0 \exists N\in\mathbb{N}: n,m\ge N\implies |x_n-x_m|<\epsilon.$$ I have tried to prove: $ | \frac{ n-sin(n)}{n} - \frac{ m…
smalllearner
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Alternate definition of Cauchy sequence

Is it true that $x_n$ is a Cauchy sequence in some Hilbert space if and only if: for all $m \in \mathbb{N}$, for all $\epsilon > 0$, there exists $N=N(m) \in \mathbb{N}$ (so the $N$ can depend on $m$) such that $$\lVert x_{n+m}-x_n \rVert \leq…
StopUsingFacebook
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Help with proofs

Could you guys help me come up with examples of sequences that follow these statements so I can understand how to do these proofs please!! 1) $\forall \epsilon > 0, \exists n > 1$ such that $q_n>1$ and $0<|q_n − q_{n+1}|< \epsilon$. 2) $\forall K >…
Maths
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Need help on monotone increasing sequence

Let $\{a_n\}_{n\in\mathbb N}$ be a monotone increasing sequence where there exists $N\in\mathbb N$ such that for any $n, m \in\mathbb N$ with $n, m \geq N$ we have that $|a_n − a_m| \leq 2$. Show that $\{a_n\}_{n\in\mathbb N}$ is convergent.
Karasuno
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Cauchy criterion approximation

I have to use Cauchy's criterion on $$\sum_{n=1}^\infty \frac{\cos{n}a}{3^n}.$$ Because I'll have the absolute value of it to prove Cauchy and cos n is from [-1,1], so it will actually be from [0,1], can I approximate the fraction to just…
Stefana
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Is $a_n=\sum^n_{k=1} \frac{1}{k}$ a cauchy sequence?

A cauchy sequence is bounded and $a_n=\sum^n_{k=1} \frac{1}{k}$ is unbounded. But because each term is smaller than the next it seems like I can find an $N$ such that the difference between all terms past $N$ is less than $\epsilon$ which would…
Tsangares
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