Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

294 questions with no upvoted or accepted answers
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6
votes
1answer
468 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 $...
6
votes
0answers
560 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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1answer
8k views

Proving convergent sequences are Cauchy sequences

Prove that if $x_n \rightarrow a, n \rightarrow \infty$ then $\{x_n\}$ is a Cauchy sequence. I believe I have found the proof as follows, wondering if there are any simpler methods or added ...
5
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0answers
620 views

Prove the space of bounded sequences is Banach

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/exam1sol.pdf Here is a proof, but I cannot fully understand why it does not give a proof that $x$ is a bounded sequence (i.e. $x$ is in the ...
5
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0answers
203 views

Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
4
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0answers
287 views

Using Cauchy's Criterion to show non-uniform convergence of series of functions.

I want to show $$\sum_{n=0}^\infty x^n$$$$x\in(-1,1)$$ does not converge uniformly using the negation of Cauchy's Criterion for uniform convergence of series of functions. Cauchy's Criterion states ...
4
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0answers
166 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
123 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
151 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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0answers
85 views

Proving that every Cauchy sequence in measure converges in measure

Let $(X,\mathcal{A},\mu)$ be a measure space and $(f_n)$ a sequence of real-valued functions on $X$ which is Cauchy in measure; that is, for any $\epsilon>0$ there exists $N\in\mathbb{N}$ such that ...
3
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0answers
60 views

Is the space $(X,d)$ complete? If not what is its completion?

Is the space $(X,d)$ complete? If not what is its completion? a) X is the set of all continuous on [0,1] functions, $$d(x, y) = \sup_{0 \le t\le1}\ t^2 |x(t) - y(t)|$$ b) $X = \{x\in C[0,1]: \sup_{0 ...
3
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0answers
77 views

If isolated points are not dense, then removing their closure leaves a space without isolated points

I have to show that for a complete metric space $Z$, if $Z\ne\overline{\operatorname{iso}\left(Z\right)}$, then $\operatorname{iso}\left(Z\setminus\overline{\operatorname{iso}\left(Z\right)}\right)=\...
3
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2answers
140 views

Prove that it is a cauchy sequence

Show that the sequence $\langle f_n\rangle$ where $$f_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots + \frac{(-1)^{n-1}}{n}$$ is a cauchy sequence. My Approach: I tried solving it by starting it ...
3
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1answer
67 views

How to prove convergence of sequence with $\epsilon$?

If we have sequence $(\frac{4^{n}+1}{5^{n}})_{n\in \mathbb{N_{0}}}$, it is easy to calculate limit of it: $\lim_{n \to \infty} = \frac{4^{n}+1}{5^{n}}=\frac{(\frac{4}{5})^{n}+\frac{1}{5^{n}}}{1}=\...
3
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0answers
101 views

Complete in $L^1$ but not $\mathrm{sup}$ norm

While learning a bit of functional analysis from an introductory book I got stumped by the following problem: Find a linear space complete in the $L^1$ norm, $||f||_1 \equiv \int_0^1 |f(t)|\ \mathrm{...
3
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1answer
95 views

Completeness of $l_1 ^\infty$

I'm trying to prove that $l_p ^\infty $ is complete for each $p\geq 1$ but only with the definition of $\varepsilon$-$N$. I know that this have been proved in other posts here but I couldn't find a ...
3
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0answers
55 views

Show that $(3x_{n}+4y_{n})$ is also Cauchy sequence.

Show that if $(x_{n})$ and $(y_{n})$ are Cauchy sequences in $X$, then the sequence $(3x_{n}+4y_{n})$ is also Cauchy sequence using the definition of a Cauchy sequence. Attempt Let $\epsilon > 0$ ...
3
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2answers
65 views

Cauchy sequence that satisfies $\|x_{k+2}-x_{k+1}\|\le\theta\|x_{k+1}-x_k\|$

Suppose the sequence $\{x_k\}_{k=1}^\infty\subset\mathbb{R}^n$ satisfies $\|x_{k+2}-x_{k+1}\|\le\theta\|x_{k+1}-x_k\|$ for all $k\ge1$, with $0<\theta<1$. Show that $\{x_k\}$ is a Cauchy ...
3
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0answers
300 views

Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
3
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0answers
168 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
3
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1answer
145 views

Prove that the sequence $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$ converges using Cauchy

I need some help with a homework question i'm having difficulty with. Here is the question: "Use the definition of cauchy sequence to prove that the series $\left(1+\frac{2}{3+1}+\frac{3}{9+1}+\cdots+...
3
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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 ...
2
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0answers
40 views

Convergence of complex sequence with gradient descent

I am looking to solve the following otimization: $\underset{{{{\bf x} \in \mathcal{X}}}}{\text{min}} \; f({\bf x})$ where ${\bf x} \in \mathbb{C}^N$ and $f({\bf x}) \in \mathbb{R}$, $f(x)$ is a ...
2
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0answers
32 views

Cauchy Sequence in $C[0,1]$ endowed with $L^2$ norm and $L^\infty$ norm

Take $V=C[0,1]$ with the usual $L^2$ norm, i.e., $\|f\|_2^2 = \int_{0}^{1}|f(\tau)|^2d\tau$. Is it complete? Consider the following sequence of functions: $$f_n(t) = \begin{cases} 0 & t &...
2
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0answers
52 views

Will a convergent Double Sequence be bounded also?

A convergent Double Sequence will be bounded also. My Attempt: I think the statement is not true. Counter Example : $a_{1n} = n$, $ a_{mn} = 1/m + 1/n$ for all $m \geq 2$ lim$_{m,n ...
2
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0answers
50 views

Proof of X is not Banach

Set $X = \lbrace u\in\mathcal{C}^2 [0,\pi]: u(0)=u(\pi)=0\rbrace$ equipped with the norm $$\Vert u \Vert = \left(\int_0^\pi (u'(x))^2\ dx + \int_0^\pi u(x)^2\ dx\right)^{1/2}$$ I want to prove that $(...
2
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0answers
52 views

Confusion about Cauchy sequences

We know that a sequence of real numbers $(x_n)_{n\ge 1}$ is a Cauchy sequence if $\forall \epsilon>0$ $\exists n(\epsilon)$ so that $|x_{n+p}-x_n|<\epsilon$ $\forall n \ge n(\epsilon)$ and $p \...
2
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0answers
57 views

How to prove convergence using cauchy sequences

Let $(a_n)$ be a monotone and bounded sequence such that $a_n \to a$. Let $(b_n)$ be defined as $b_n = (a_1 + a_2 + ... + a_n)/n$. I know $(b_n)$ is monotone and bounded, but how do I prove that $b_n \...
2
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0answers
66 views

Proof verification of $x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$ diverges by the negation of Cauchy

Let $\{x_n\}$ denote a sequence: $$ x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2} $$ Show that $\{x_n\}$ diverges using the negation of Cauchy criterion. I would like to kindly request a ...
2
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0answers
72 views

Concerning the existence of a divergent monotone sequence with a Cauchy subsequence.

Is my argument correct? Proposition. There is no divergent monotone sequence with a Cauchy subsequence. Proof. Assume that we have a divergent monotone sequence $(a_n)$ with a Cauchy sequence $(a_{...
2
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0answers
78 views

Convergent sequences in topological groups

I'm trying to solve an exercise regarding sequences on topological groups, Consider $G$ a topological group, i have to prove that the class of convergent ( to $0$ ) sequences is closed in the class of ...
2
votes
1answer
37 views

Show that $(B(X), D)$ is complete.

Exercise: Let $\Omega$ be a nonempty set. Let $B(\Omega)$ be collection of bounded real-valued functions on $\Omega$. Define $D:B(\Omega)\times B(\Omega)\to[0,\infty)$ by $$D(f,g) = \sup_{w\in\Omega}\...
2
votes
3answers
207 views

Cauchy criterion for sequences convergence

The task is to find out if the following sequence converges: $x_n = 1 + \dfrac{\sin(1)}{1^2} + \dfrac{\sin(2)}{2^2} + \ldots + \dfrac{\sin(n)}{n^2} $ I don't even know what to do, can you help me in ...
2
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0answers
146 views

Let $\{x_n\}$ be a Cauchy sequence of rational numbers. Define a new sequence $\{y_n\}$ by $y_n = (x_n)(x_{n+1})$. Show that $\{y_n\}$ is a CS.

What I am thinking so far is following: Construct another sequence $\{b_n\}$ such that $b_1=x_2, b_2=x_3, \ldots, b_n=x_{n+1}.$ Since $\{x_n\}$ is a Cauchy sequence, $\{b_n\}$, as a subsequence of $\{...
2
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0answers
21 views

Prove that ${\xi_i}$ is complete system in $l_2$

Let ${\{x_i\}} \subset \mathbb{C}$, $x_i \neq 0$, $x_i \rightarrow 0$, $|x_i| < 1$. $\xi_i$ = {$x^k_i$}$_{k\ge0} \Rightarrow$ {$\xi_i$} is complete system of sequences in $l_2$. I should prove ...
2
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0answers
156 views

Patterns appearing in irrational approximation graphs

I'd like to know more about some patterns I found in graphs corresponding to irrational numbers. Here's the graph for $\sqrt 2$ for example First, I'll try to explain most naturally the function that ...
2
votes
1answer
96 views

What's the diagonal Cauchy subsequence (to show totally bounded if and only if every sequence has a Cauchy subsequence)?

I read a textbook showing a subset of a normed linear space is totally bounded if and only if every sequence in it has a Cauchy subsequence. It proves as follows: $\Rightarrow$ Let $(x_n)$ be an ...
2
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0answers
266 views

Cauchy sequence in $\mathbb{Q}_p$

Let $\left\{a_i\right\},\left\{b_i\right\}$ be two Cauchy sequences with respect to norm $|\cdot |_p$ on $\mathbb{Q},$ that is, $a_i,b_i\in \mathbb{Q},$ and $$\forall \ \varepsilon>0,\exists N\in \...
2
votes
1answer
81 views

Completion of a module is equivalent to Cauchy sequence criterion

Suppose $M$ is an $A$-module where $A$ is a commutative ring with $1$. $M$ has the $I$- adic topology where $I$ is an ideal of $A$. We define the completion of $M$, $\hat M:=\varprojlim M/I^nM$. Then ...
2
votes
0answers
138 views

Lipschitz conditions proof

Let $M_K$ be the set of all function $f$ in $C_{[a,b]}$ satisfying a Lipschitz condition, i.e., the set of all $f$ such that $$|f(t_1)-f(t_2)|\leq K|t_1-t_2|$$ for all $t_1,t_2 \in [a,b]$ where $K$ ...
2
votes
0answers
45 views

Constructing a special kind of sequence in an infinite dimensional normed linear space

Show that every infinite dimensional normed linear space contains a sequence $(x_n)$ such that $\|x_n\|=1$ $\forall$ and $\|x_m-x_n\| \geq 1$ for all $m ,n \in \mathbb{N}$ and $m \neq n$. I tried to ...
2
votes
2answers
800 views

Convergent sequences in a metric space

Consider the set N of natural numbers with the metric $$d(m,n) = \left\lvert{\frac{1}{m}−\frac{1}{n}}\right\rvert \; \mathit{n, m ∈ \mathbb{N}} $$ Describe all convergent sequences in this metric ...
2
votes
0answers
33 views

If set A is complete, prove that A is closed(proof-explanation)

I marked a place, which confuses me: So, how do we know, that if A is complete it means, that sequence ${x_n}$ converges to a point, that belongs to A?
2
votes
0answers
155 views

Prove that a sequence is Cauchy in an inner product space.

I need to show that the space $H=B(\mathbb{R},\mathbb{C})$ of complex bounded real-valued functions equipped with the inner product $$\langle f,g\rangle=\int_{\mathbb{R}}\frac{f(x)\overline{g(x)}}{1+...
2
votes
0answers
569 views

Proof that the metric space of convergent sequences is complete

Let $c_{0}$ be the space of real-valued sequences $\{x_{n}\}$ which converge to zero, equipped with the metric $d(\{x_{n}\}, \{y_{n}\}) = \sup\{|x_{n} – y_{n}|: n \in \mathbb{N}\}$. Let $e_{k}$ denote ...
2
votes
1answer
417 views

Compact subset of the space of all bounded sequences of real numbers

Let $X$ be the metric space of bounded sequences of real numbers $ x = (x_n) $ with the metric $ d(x,y) = \sup_n |x_n - y_n| $. Show that the set $$ Y = \{ x = (x_n) \in X \mid |x_n| \leq c_n = \text{...
2
votes
2answers
36 views

Basic Analysis involving Cauchy Sequences

Is $s_n$ a Cauchy sequence if we only assume that $|s_{n+1} - s_n|\lt \frac{1}{n}$ for all $n\in \Bbb{N}$ My original question was poorly worded, hopefully this make more sense. I get that it is not ...
2
votes
0answers
154 views

A metric space is compact iff every closed ball is compact

Is this true? I think I have a counter example. If we consider the set $(\mathbb{N},d)$, where $d$ is the discrete metric, then every subset closed ball is compact, but since $\mathbb{N}$ is infinite $...
2
votes
0answers
69 views

Interpreting $\sqrt{2}$

My apologies if this question is somewhat vague or too broad. Analytically, the fact that $\sqrt{2}$ is no trivial fact. It requires some kind of completion of the rational numbers, e.g. by adjoining ...