Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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171 views

Proof review - (lack of rigour?) Convergent sequence iff Cauchy without Bolzano-Weierstrass

I am currently trying to improve my skills doing epsilon-delta proves and I just attempted the following one. Since I'm such a newbie in calculus I would like to improve learning form my mistakes (...
4
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0answers
124 views

Showing a sequence is Cauchy; loss of generality?

The exercise is as follows; Show that the sequence $$(a_n) = \left(\frac{(-1)^n}{\sqrt{n}}\right)_{n \in \Bbb N}$$ is a Cauchy Sequence. Solution: Let $m > n.$ Since we are trying to show ...
4
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0answers
155 views

My proof of: Every convergent real sequence is a Cauchy sequence.

Is my proof correct? Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every $\...
4
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1answer
73 views

Fixed point question with convergence

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ is $C^1$ and $1$ to $1$ and there exists a strict increasing sequence $t_{n} \in \mathbb{N}$ s.t $f^{t_{n}}(x) \to p$ for all $x$ as $n\to \infty$ (composition $...
3
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5answers
1k views

If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
3
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2answers
4k views

Is the set of integers Cauchy complete?

http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
3
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1answer
326 views

Why we define the completeness of a space by the converge of a Cauchy sequence rather than a normal sequence?

The intuition of the completeness to me is that the limit of any sequence converges to the point inside the set itself. But why we define a set to be complete as any Cauchy sequence converge into the ...
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3answers
1k views

To prove a sequence is Cauchy [duplicate]

I have a sequence: $ a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } } $ , it repeats $n$-times. and i have to prove that it is a Cauchy's sequence. So i did this: As one theorem says that every ...
3
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4answers
112 views

The sequence $x_{n+1}=ax_{n}+b $ converges to where?

$$a,b \in \mathbb R , \ 0\lt a\lt 1 . $$ Define the sequence $$x_{n+1}=ax_{n}+b \text{ for } n\ge0\ .$$ Then for a given $\ \ x_0\ \ $ , does this sequence converge? And if it does, to ...
3
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2answers
200 views

Determine if this specific sequence is a Cauchy sequence

I have the following sequence: $$a_n =\sum_{k = 1}^n (-1)^{b_k} {1\over k^2} $$ And the hint is that I have to prove that: $$ {1\over k^2} < {1\over k-1} - {1\over k} $$ So assuming $m>n$, I ...
3
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2answers
1k views

My Proof: Every convergent sequence is a Cauchy sequence.

Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every $\varepsilon \in \mathbb{R}$ ...
3
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2answers
51 views

Existence of $a_k$ such that $\sum_k a_kb_k<\infty$ and $\sum_k a_k=\infty$ given $b_k\to 0$

I was working with a problem from functional analysis. I reduced the problem to the following problem: Let $b_k>0$ be a decreasing sequence converging to $0$. Does there exist a non-negative ...
3
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2answers
72 views

Does $\sum_k \frac{a_k-a_{k-1}}{a_k^2}$ converge if $a_k \uparrow \infty$?

Let $a_n$ be a sequence of strictly increasing postive numbers such that $a_n \uparrow \infty$. Does $$\sum_{k=1}^\infty \frac{a_k-a_{k-1}}{a_k^2}$$ converge? My guess is that it converges. I tried ...
3
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3answers
2k views

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
3
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3answers
10k views

Confused with proof that all Cauchy sequences of real numbers converge.

First the textbook proves that all Cauchy sequences are bounded, and so have a convergent subsequence, $\{a_{n_{k}}\}$ that converges to a limit, say $L$. Now we use this to prove that all Cauchy ...
3
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3answers
110 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
3
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1answer
79 views

If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$

If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$ I feel like answer should be NO but I am unable to find such an exapmle. Can anybody please help ...
3
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4answers
196 views

Show a sequence such that $\lim_{\ N \to \infty} \sum_{n=1}^{N} \lvert a_n-a_{n+1}\rvert< \infty$, is Cauchy

Attempt. Rewriting this we have, $$\sum_{n=1}^{\infty} \lvert a_n-a_{n+1}\rvert< \infty \,\,\,\Longrightarrow\,\,\, \exists N \in \mathbb{N}\ \ s.t,\ \ \sum_{n \geq N}^{\infty} \lvert a_n-a_{n+1}\...
3
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3answers
4k views

Prove a Sequence is Cauchy

Prove that sequence {(2n+1)/n} is Cauchy. I understand the definition of a Cauchy sequence; however, I'm not sure how to find the necessary value of N to satisfy the prove. I know that you can ...
3
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2answers
140 views

Why can't completeness be defined on topological spaces without using metrics?

I have heard it said that completeness is a not a property of topological spaces, only a property of metric spaces (or topological groups), because Cauchy sequences require a metric to define them, ...
3
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2answers
51 views

How unboundedness of $\|T_n\|$ will imply that X is not complete?

I have to show that The normed space $X$ of all polynomials with norm defined by $$\|x\|=\max\vert\alpha_j\vert$$ ($\alpha_0,\alpha_1,...$the coefficients of $x$) " is not complete using ...
3
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4answers
773 views

Why is the sequence $x(n) = \log n$ **not** Cauchy?

I read in the book Applied Analysis by Hunter and Nachtergale that the sequence $x(n)=\log(n)$ is not Cauchy since $\log(n)\to\infty$ But that seems to be irrelevant to the definition of a Cauchy ...
3
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3answers
77 views

Approximating the limit of a Cauchy sequence in a Banach space

Let $E$ be a Banach space and consider a sequence $(x_n)_n$ in $E$ satisfying the following condition: $$||x_n-x_{n-1}||\leq 3^{-n}\mbox{ for all }n\in\mathbb{N}.$$ Clearly $(x_n)_n$ is a Cauchy ...
3
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5answers
183 views

Proving that the sequence $\{\frac{3n+5}{2n+6}\}$ is Cauchy.

I'm not quite sure how to tackle these kinds of questions in general, but I tried something that I thought could be right. Hoping to be steered in the right direction here! Let $\{\frac{3n+5}{2n+6}\...
3
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2answers
75 views

$(X,d)$ be a complete metric space, and $d(x_{n},x_{n+1}) \leq \frac{1}{n^2} , d(y_{n},y_{n+1}) \leq \frac{1}{n}$ .

Let $(X,d)$ be a complete metric space, and $x_{n} , y_{n}$ be sequences in $X$ such that $d(x_{n},x_{n+1}) \leq \frac{1}{n^2} , d(y_{n},y_{n+1}) \leq \frac{1}{n}$ then does $(x_{n}) , (y_{n})$ ...
3
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3answers
95 views

Prove that two sequences converge or are Cauchy

I have the two following implications to prove. The first is this: Assume that $\forall k\in \mathbb{N}, \{f_n(k)\}_{n=1}^\infty \subset \mathbb{R}$ converges in $\mathbb{R}$. Then prove that $\{f_n\}...
3
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2answers
418 views

How to prove that these are Cauchy sequences?

Let $x_1,x_2,x_3,\dots$ is a non-decreasing and $y_1,y_2,y_3,\dots$ is a non-increasing sequence, and they are real sequences. If $|x_n-y_n|\le\frac{|x_1-y_1|}{2^n}$ for each $n\in \Bbb N$, then show ...
3
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2answers
1k views

Cauchy sequence in compact metric space converges; incorrect proof?

I'm self-studying real analysis, and I've been trying to prove the following statement: If $X$ is a compact metric space and if $\{p_n\}$ is a Cauchy sequence in $X$, then $\{p_n\}$ converges to some ...
3
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3answers
151 views

How to prove that $\frac{1+4n^2}{2+2n^2}$ is a cauchy sequence?

I know that the following sequence is a $$\frac{1 + 4n^2}{2+2n^2}$$ How can I show that it is a cauchy sequence - well it has to cause its convergent but I want to understand it with the cauchy ...
3
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4answers
994 views

Recursive sequence with square root

I came across this (cool) question this weekend Find the limit of the following sequence as $n$ approaches infinity. $x_1 = 1$ and $x_{n+1} = \sqrt{x_n^2+\frac{1}{2}^n}$ I had two questions about it....
3
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2answers
1k views

Convergent Sequence and Cauchy Criterion- Counter Example

Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition: $$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}} \ \ \ for\ all\ n=1,2,3,...$$ Part (1): Prove that the sequence $\...
3
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3answers
43 views

Cauchy sequences of rationals with limit irrational: natural, or geometric examples

As we know, real numbers are constructed by filling up gaps between rationals by the limits of all Cauchy sequences of rationals. Q. What are examples of sequence of rationals such that its easy ...
3
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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
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3answers
144 views

Cauchy sequence $x_n=\sqrt{a+x_{n-1}}$

I have to show that this sequence $$ x_n=\sqrt{a+x_{n-1}} \hbox{ with } x_1=\sqrt{a} $$ is a Cauchy sequence for every $a>0$. I have done the following calculations: $$ \left| x_{n+2}-x_{n+1} \...
3
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1answer
1k views

$C[0,1]$ with $L^1$ norm is not Banach space.

I want to check that $(C[0,1],∥⋅∥_1)$ is not a Banach space, where $\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x$.I took $(f_n)_{n \geq 1}$ a sequence in $C[0,1]$ given by:$f_n: [0,1] \rightarrow \mathbb{R}, \...
3
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3answers
415 views

$\{a_n\}$ is a Cauchy sequence, if $a_{n+2} = \frac{a_n + a_{n+1}}{2}$

Suppose that the sequence $\{a_n\}$ satisies the relation $$ a_{n+2} = \frac{a_n + a_{n+1}}{2}, $$ for all $n \in \mathbb{N}_{+}$ Prove that $\{a_n\}$ is a Cauchy sequence ...
3
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2answers
434 views

Uncountably many equivalent Cauchy sequence?

RTP There exists uncountably many Cauchy sequence of rationals that are equivalent. I am trying to solve the above question, and my understanding is that $\Bbb R$ is a set of equivalent classes of ...
3
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3answers
51 views

Given a Cauchy sequence $a_n$, show that $\sqrt{a_n}$ is Cauchy when $a_n>0$ for all $n$.

We have a sequence $a_n$, that is Cauchy and every term is positive. How do I find that $\sqrt{a_n}$ is also Cauchy? I have seen a similar question posted but in that question $a_n>1$ so it is not ...
3
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2answers
70 views

Proof verification that $\{x_n\} = 0,\underbrace{77\dots 7}_{\text{n times}}$ is a Cauchy sequence.

Given a sequence $\{x_n\}$: $$ x_n = 0,\underbrace{77\dots 7}_{\text n\ times} $$ Prove that $\{x_n\}$ is a Cauchy sequence. Recall the definition of a fundamental sequence: $$ x_n\ \text{is ...
3
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2answers
84 views

Show that $a_n = \frac{n}{2n+1}+\frac{1}{n^3}$ is a Cauchy Sequence

I want to show that $$a_n = \frac{n}{2n+1} + \frac{1}{n^3}$$ is a Cauchy sequence. My attempt: $$|a_m-a_n|=|(\frac{m}{2m+1}-\frac{n}{2n+1}+\frac{1}{m^3}-\frac{1}{n^3})|$$ $$\leq\frac{m}{2m+1}+\...
3
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1answer
4k views

Prove that a Cauchy sequence is convergent

I need help understanding this proof that a Cauchy sequence is convergent. Let $(a_n)_n$ be a Cauchy sequence. Let's prove that $(a_n)_n$ is bounded. In the definition of Cauchy sequence: $$(\...
3
votes
1answer
491 views

Divergent sequence with decreasing function

Let $f: \mathbb{R} \to (0,\infty)$ be a decreasing function. Define a sequence $(a_n)$ by $a_1=1 $ and $a_{n+1}=a_n+f(a_n)$ for every $n\ge 1$. Prove that $(a_n) \to \infty$. I have tried by ...
3
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2answers
714 views

Sum of the distance between consecutive terms implies the sequence is Cauchy.

If a sequence $(x_n)_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum_{n\geq 1} ||x_n-x_{n+1}||<\infty$, show that it is Cauchy. This isn't a complete answer, but here's my train of thought. ...
3
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2answers
1k views

Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy ...
3
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4answers
477 views

Alternate proof that a sequence is Cauchy

Is it sufficient to show that for any $\epsilon > 0$ that there exists $N$ such that if $n$ is greater than or equal to $N$, then $d(s_n, s_{n+1})$ is less than $\epsilon$ to prove that a sequence $...
3
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2answers
114 views

How to show that $x_n=(1+\frac{1}{n})^n$ is a cauchy sequence in $\mathbb Q$. [duplicate]

How to show that $x_n=(1+\frac{1}{n})^n$ is a cauchy sequence in $\mathbb Q$. I am trying to use binomial theorem to expand $(1+\frac{1}{n})^n$ but it is not coming.
3
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4answers
39 views

Convergence of Cauchy's sequence

I understood that every convergent sequence is a Cauchy sequence. It seems that the converse is not necessarily true. An example given is the sequence $\{x_n\}$, where $x_n = (0.1)^n$ is a Cauchy ...
3
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3answers
76 views

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
3
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2answers
90 views

How to show that $\sin(n)$ does not converge ONLY by using Cauchy's criterion?

I know this question has been asked before... I went through all of the questions of this sort and none of them had an answer using Cauchy's criterion. I know that $\sin(n)$ does not converge and I ...
3
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1answer
159 views

Question about Cauchy Sequence proof for $\sum_{i=1}^\infty \frac{1}{i^2}$

I was looking through a proof that the sum of $$\sum_{i=1}^\infty \frac{1}{i^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}...$$ is convergent. I know there are probably other ways to prove this, ...