Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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prove that the unit circle $x^2+y^2=1$ is a closed set

I need to prove that the unit circle $x^2+y^2=1$ is a closed set in $\mathbb{R}^2$ is closed using convergent sequence method.
6
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2answers
807 views

Is every Cauchy sequence in a non-complete metric space convergent?

A metric space $X$ is called complete if every Cauchy sequence in $X$ has a limit in $X$. For a non-complete metric space $X$, can we say that every Cauchy sequence is convergent? (even though the ...
6
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1answer
614 views

challenge problem: show this sequence is convergent.

had this difficult question from a textbook, and I haven't been able to figure out the solution. say we have a sequence of bounded real numbers $a_n$ such that $2a_n \leq a_{n-1} + a_{n+1} \forall n\...
6
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3answers
68 views

Does convergence for Cauchy sequence fail only when the limit is not in the domain?

I am trying to understand how important is the distinction between Cauchy sequences and convergent sequences in normed vector spaces $E$. So far I have only come across examples where the Cauchy ...
6
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1answer
2k views

If $f$ takes Cauchy sequence to Cauchy sequence then $f$ is continuous [duplicate]

If $f:X\to Y$ takes Cauchy sequence to Cauchy sequence then prove that $f$ is a continuous function. Let $x_n$ be a sequence in $X$ such that $x_n\to x\implies x_n$ is Cauchy $\implies f(x_n)$ is ...
6
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1answer
316 views

How can one check if a Cauchy-sequence converges in the rationals?

Let $(x_k)$ be a sequence in $\mathbb Q$ such that $x_k=\sum\limits_{n=1}^{k}\frac{1}{10^{n^2}}$ for all $k\geq 1$. It can be easily seen that this sequence is bounded and Cauchy. But does it ...
6
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3answers
178 views

Is this a valid convergence test (for sequences)?

Let $(x_n)$ be a bounded sequence (from above by $M$) such that $\lim_{n\to\infty}\frac{x_{n+1}}{x_n}=1$. Does $(x_n)$ converge? My first idea was to prove the sequence is Cauchy: Given $\epsilon\gt0$...
6
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2answers
576 views

What's wrong with the classical Cauchy construction of the reals?

I am reading Bishop's "Constructive Analysis" and he says that defining a real number to just be an equivalence class of Cauchy sequences of rationals would be wrong. Why is that?
6
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2answers
101 views

Bounded sequence $\{a_n\}_n$ such that $a_n < \frac{a_{n−1} + a_{n+1}}{2}$. Is $\{a_n\}_n$ convergent?

Let $\{a_n\}_n$ be a sequence of numbers in the interval $(0, 1)$ with the property that $$a_n < \frac{a_{n−1} + a_{n+1}}{2}$$ for all $n = 2, 3, 4,\dots$. Show that this sequence is convergent....
6
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1answer
175 views

showing $a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$ is not Cauchy

My gut telling me that the following sequence is not Cauchy, but I don't know how to show that. $$a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$$
6
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Need to prove $f$ continuous at $x_0$ iff for every monotonic sequence $(x_n)$ converging to $x_0$ we have $\lim f(x_n)=f(x_0)$

This was a problem that the Professor went over in class, but I am having trouble understanding and finishing the proof. The full question is: $f:I \rightarrow \mathbb R$ is continuous at $x_0 \in I$ ...
6
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2answers
100 views

Show that the series $\sum \frac{\sin \left(\frac{\left( 3-4n \right)\pi }{6}\right) }{2^{n}}$ converges?

Using the addition formula for the sine function I have managed to reduce this to a simpler form: $$\sum \frac{\cos \frac{2n\pi }{3}}{2^{n}}$$ It is obvious here that it passes the n-th term ...
6
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1answer
941 views

Cauchy sequences. Show that $(x_n)$ is Cauchy.

Let $(x_n)$ and $(y_n)$ be sequences such that $\lim y_n = 0$. Suppose that for all $k \in \Bbb N$ and all $m ≥ k$ we have $|x_m − x_k| ≤ y_k$. Show that $(x_n)$ is Cauchy. I need a little guidance ...
6
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3answers
453 views

If $|x_{n+1} - x_n| < \epsilon,$ is $(x_n) $ a Cauchy sequence?

Suppose that $\, \forall \epsilon >0, \, \exists N, \,$ such that $$|x_{n+1} - x_n| < \epsilon\, \quad \forall n \geq N.$$ Is the sequence $(x_n)$ a Cauchy sequence? If so, prove it; if not,...
6
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2answers
4k views

Show that every monotonic increasing and bounded sequence is Cauchy.

The title is kind of misleading because the task actually to show Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy without knowing that: Every bounded non-empty ...
6
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1answer
2k views

Show that the image of a complete metric space under a continuous map is also complete given an additional condition.

This is a problem from revision material for a functional analysis class. Let $(X,d)$ and $(C,p)$ be two metric spaces and let $f:X\rightarrow C$ be a continuous function with $f(X)=C$. Assuming $(...
6
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1answer
3k views

Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences ...
6
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1answer
184 views

proving that $S_n$ is Cauchy.

$$S_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... + \frac{(-1)^{n+1}}{2n-1} $$ Show that $(S_n)$ is a Cauchy sequence and hence that it converges to limit $L$. Show that $\frac{2}{3} < L &...
6
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1answer
216 views

Proving a sequence is Cauchy given some qualities about the sequence

I've got a sequence $x_n$ such that I've proved $b\leq x_n \leq c$, and $|x_{n+1}-x_{n}|\leq \frac{4}{9}|x_n-x_{n-1}|$ However I'm not very familiar with Cauchy sequences, so I don't know how to ...
6
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0answers
103 views

Prove: $\lim\limits _{n\rightarrow\infty} \left( ((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}} \right)=\frac{1}{e}$ [duplicate]

I see this solution in a book but I don't understand it. Which theorems were used? I know how I solve this limit but this solution in book as following: $$\lim\limits_{n \rightarrow \infty} \left(...
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1answer
474 views

Prob. 11, Chap. 4 in Baby Rudin: uniformly continuous extension from a dense subset to the entire space

Here is Prob. 11, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $...
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0answers
575 views

Equivalent definition of Cauchy sequence

A sequence $x_i$ is Cauchy if for all $r>0$, there exists $n$ s.t. $i,j\geq n$ implies $d(x_i,x_j)<r$. My question is, is it equivalent to define Cauchy as follows? $x_i$ is Cauchy if for all $...
5
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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
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2answers
2k views

Is it possible to define Cauchy sequences in a topological space?

I know that we can define Cauchy sequences in topological vector spaces. How about in general topological spaces? Is it possible to define a Cauchy sequence in general topological spaces?
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2answers
5k views

Metric space is totally bounded iff every sequence has Cauchy subsequence

Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. I think I proved the Cauchy subsequence part: Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be ...
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5answers
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I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.

My understanding from the definition in my book (Rudin) is this. A seq. $\{p_n\}$ in a metric space $X$ (I only really know $\mathbb R^k$) is said to be a Cauchy sequence if for any given $\epsilon ...
5
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2answers
836 views

Proving a contraction mapping is a Cauchy sequence

Let $\phi(x):[a,b]\rightarrow [a,b]$ be a continuous function. Show that if $\phi(x)$ is a contraction mapping on $[a,b]$ then the sequence $\{x^{(k)}\}$ defined by $x^{(k+1)} = \phi(x^{(k)})$ is a ...
5
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3answers
2k views

How do I find the limit of the sequence $a_n=\frac{n\cos(n)}{n^2+1}$ and prove it is a Cauchy sequence?

I need to study the limit behavior of $a_n=\frac{n\cos(n)}{n^2+1}$, which can be written as $\frac{n}{n^2+1}\cos(n).$ I knew that it wasn't going to be monotone because $cos(n)$ oscillates between -1 ...
5
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2answers
269 views

Convergence of $a_{n+2} = \sqrt{a_n} + \sqrt{a_{n+1}}$ [duplicate]

Let $a_1$ and $a_2$ be positive numbers and suppose that the sequence {$a_n$} is defined recursively by $a_{n+2} = √a_n + √a_{n+1}$. Show that this sequence is convergent. So, I have been able to ...
5
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1answer
258 views

For $f:X\rightarrow Y,$ if $f$ is continuous and $X$ is a complete metric space, does $f$ preserve Cauchy sequences?

For $f:(X,d)\rightarrow (Y,ρ),$ if $f$ is continuous and $(X,d)$ is a complete metric space, does $f$ preserve Cauchy sequences (i.e. $(x_n)$ is Cauchy in $X$ $\Rightarrow$ $(f(x_n))$ is Cauchy in $Y$)...
5
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4answers
211 views

How to determine what kind of Cauchy sequences lie in a given space?

I understand the main principles of Cauchy sequences and metric spaces, but I have a particular question about determining whether or not a space is a complete space. If a space has all cauchy ...
5
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2answers
138 views

Intuition for non-convergence of Cauchy sequence in $\mathbb{Q}$

Suppose we were standing on the rational line at the point 3. Then we took a step to the point 3.1, then to 3.14, etc. (Cauchy sequence of decimal approximations of $\pi$). Suppose, also, that it ...
5
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3answers
297 views

How to prove divergence elementarily

I have this problem: give an example of a real sequence $\;\{a_n\}\;$ with $$\lim_{n\to\infty}\left(a_{n+1}-a_n\right)=0\;,\;\;\text{but}\;\;\lim_{n\to\infty}a_n\;\;\text{doesn't exist finitely}$$ ...
5
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2answers
535 views

Is every closure of a metric space a completion?

I know that every completion is a closure of a metric space, since every convergent sequence is cauchy and and the limit of that sequence will exist within the completion. At the same time, from my ...
5
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1answer
163 views

If two metric spaces are Cauchy-equivalent, then are they topologically equivalent?

If two metrics $d_i$ on the same set $X$ have the same Cauchy sequences (i.e. if a sequence is Cauchy for the first metric, it is also Cauchy for the other one and vice versa), does this imply that ...
5
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1answer
802 views

Does a continuous function take Cauchy sequences to Cauchy sequences?

Let $(X,d)$ be a metric space. A function $f:X\rightarrow X$ maps Cauchy sequences to Cauchy sequences. Then if $x_n\rightarrow x$, then $\{x_n\}$ is Cauchy, implying $\{f(x_n)\}$ is Cauchy. But if $X$...
5
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2answers
121 views

Can we define sum and product of two irrational numbers using Cauchy sequences of their simple continued fraction convergents?

There is a lot of questions about sum and product of irrationals here, so I hope you'll bear with me. Simple continued fraction is a very convenient way to represent any number since every real ...
5
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1answer
1k views

A linear operator is continuous if and only if it maps cauchy sequences to cauchy sequences

Let $A$ and $B$ be seminormed spaces, then I want to show that a linear operator $T: A \rightarrow B$ is continuous if and only if it maps cauchy sequences to cauchy sequences. The direction "$T$ ...
5
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2answers
404 views

Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
5
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1answer
594 views

Prove that there exists a Cauchy sequence, compact metric space, topology of pointwise convergence

Given a compact metric space $(X,d)$, we consider $Iso(X,d)$ with metric $\rho$ such that $\lim _{n \rightarrow \infty} \rho(h_n, h) =0 \iff \forall x \in X: \lim _{n \rightarrow \infty} d(h_n(x), h(...
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2answers
159 views

Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy

Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy. My work : Need to show that for every $\epsilon \gt 0$ there exist $N$ such that $n,m\ge N \...
5
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1answer
69 views

$|x_{n + 1} - x_n| < \frac{1}{2^n} \Rightarrow (x_n)$ is Cauchy [duplicate]

Let $(x_n)$ be a real sequence with the property that for all $n \in \mathbb{N}$, $$|x_{n + 1} - x_n| < \frac{1}{2^n}$$ I want to show, using the definition of a Cauchy sequence, that $(x_n)$ must ...
5
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2answers
161 views

Convergent $\implies$ Cauchy.

I'm trying to follow a proof that if a sequence is convergent, it is necessarily Cauchy. I think I understand the proof, but I want to be sure that I'm wording the logic correctly. For that reason, ...
5
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1answer
138 views

If $a_1 + 2a_2, a_2 + 2a_3, a_3 + 2a_4 \dots$ converges, prove $a_1, a_2, \dots$ converges

Let $a_1, a_2, \dots$ be a sequence of real numbers such that the sequence $a_1 + 2a_2, a_2 + 2a_3, a_3 + 2a_4 \dots$ converges. Prove that the sequence $a_1, a_2, \dots$ must also converge. I ...
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1answer
139 views

Prob. 2, Sec. 1.4 in Kreyszig's Functional Analysis Book: Any Cauchy sequence with a convergent subsequence converges

Here is Prob. 2, Sec. 1.4 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig. If $\left( x_n \right)$ is Cauchy and has a convergent subsequence, say, $x_{n_k} \to x$...
5
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4answers
832 views

If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy.

If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy. Proof: $|a_{m_1}-a_{n_1}|\lt \epsilon_1$ and $|b_{m_2} - b_{n_2}|\lt \epsilon_2$ Then take $m_3=\max(m_1,m_2),n_3=\max(n_1,n_2)...
5
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1answer
563 views

Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $...
5
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2answers
66 views

Showing that $X= \left\{ u \in C\bigl([0, \infty);E\bigr) \mid \sup_{t \geq 0} e^{-kt} \|u(t)\| < \infty \right\}$ is Banach.

I am working on my degree thesis studying the Hille Yosida Theorem and as a motive to it, I want to present the Cauchy-Lipschitz-Picard Theorem. Haim Brezis highlights the path for a very ...
5
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1answer
87 views

Is the notion of Cauchy sequences definable in a bornological topological space?

Being a Cauchy sequence is not a topological property, i.e. two metrics can induce the same topology and yet a sequence which is Cauchy in one may not be Cauchy in the other. It is a uniform property ...
5
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1answer
198 views

Uniform convergence in convex set

I have some questions about the following theorem and it's proof. Theorem. Let $X\subset \mathbb R^n$, open, convex, bounded and $f_n:X \to \mathbb R^m$ differentiable.And let's consider the ...