Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

1
vote
3answers
77 views

Does Cauchy Completeness imply the Heine-Borel theorem generally?

I've been working through some reverse math with the completeness definitions of a metric space. More over, I've learned that in a metric space X that is ordered, The Least Upper Bound Property, ...
0
votes
1answer
59 views

If $\sum\limits^{\infty}_{n=1}|x_n|^2<\infty $ then $\{x_n\}$ is a Cauchy sequence

Let $\{x_n\}$ be a sequence of real numbers. How do I prove that if $\sum_{n=1}^\infty|x_n|^2<\infty$ then $\{x_n\}$ is a Cauchy sequence ? Is the converse true? How do I prove this? For the ...
0
votes
1answer
39 views

Integral Approximation to Infinite sum Vs Cauchy's first theorem

$$ \lim_{n\to \infty}\sum_{i=0}^n 1/(3*n +i) $$ . After applying cauchy's first theorem on pints, I get the answer as 1/4 , but after expressing the above sum as a definite integral I get the answer ...
0
votes
0answers
20 views

Prove that $\preccurlyeq$ is a linear ordering over the set of all equivalence classes of Cauchy sequences of rationals

Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an equivalence relation $\sim$ on $\mathcal{C}$ by $$(a_n) \sim (b_n) \iff \forall \epsilon >0, \exists N, \forall n>N: |...
0
votes
0answers
27 views

Prove that $(a_n) \preccurlyeq_1 (b_n) \iff (a_n) \preccurlyeq_2 (b_n)$ or $(a_n) \approx (b_n)$ for Cauchy sequences

Does my attempt look fine or contain logical flaws/gaps? To prove $\langle a_n \rangle \preccurlyeq_1 \langle b_n \rangle \Longrightarrow \langle a_n \rangle \preccurlyeq_2 \langle b_n \rangle$ or $\...
0
votes
1answer
37 views

Proof a Cauchy sequence.

Given is a sequence $(a_{n})_{n \in N}$ and is defined as $a_{n+1} - a_{n} = 1/2(a_{n-1} -a_{n})$. I have to proof that this is a Chauchy sequence, but i don't know how to start. I know that a Cauchy ...
4
votes
1answer
55 views

Let $(a_n)$ and $(b_n)$ be Cauchy sequences of rationals. Then $(a_nb_n)$ is Cauchy sequence

Let $(a_n)$ and $(b_n)$ be Cauchy sequences of rationals. Then $(a_nb_n)$ is a Cauchy sequence. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank ...
1
vote
1answer
33 views

Relation $(a_n)<(b_n)$ iff for every $\epsilon>0$, $a_n-b_n<\epsilon$ for all $n$ large enough, defines a total order $<$ on Cauchy sequences

Honestly, I figured out the idea to prove this theorem without much difficulty. But I found very hard to formalize it into a rigorous proof and it takes me three days to do it. Could you please have ...
1
vote
1answer
25 views

How to check convergence of sequence in complete metric space.

Let $\{x_n\}$ and $\{y_n\}$ be two sequences in a complete metric space $(X,d)$ such that $d(x_n,x_{n+1})\le\frac{1}{n^2}$ $d(y_n,y_{n+1})\le \frac{1}{n}$ , for all $n\in \mathbb{N}.$ ...
0
votes
0answers
28 views

For metric space $C^0([0,1])$, a sequence of functions ${x^n}$ is Cauchy in $||-||_1$ norm but not in infinite norm

In my text book, it is stated that for a sequence of functions ${x^n}$ in a metric space $C^0([0,1])$, it is Cauchy in $||-||_1$ norm but not in $||-||_\infty$ norm because that would lead to a ...
1
vote
0answers
60 views

Infinitely Nested Infinite Series / Infinite Composition of Series

Is there any documentation about a series like this? What is it called? Does it have a value? I have tried several searches and couldn't find anything close to this. Any guidance would be ...
0
votes
1answer
26 views

Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Let $(s_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}$ such that for all $n$: $0 < s_n \leq \frac{1}{n} $. Let $p>1$. How to show that the space of sequences $ l^p_s := \{x \in \mathbb{C}^\mathbb{...
1
vote
1answer
79 views

Prove convergence and find limit of recursive sequence [duplicate]

Letting $a_{1}=2$ I have a recursive sequence defined as follows: $$ a_{n+1} = \frac{a_{n}}{2} + \frac{5}{a_{n}} \ \ \forall n \geq1$$ How can I prove that the sequence {$a_{n}$} converges and also ...
0
votes
2answers
46 views

Limit of $a_n=(1-\frac13)^2\cdot(1-\frac16)^2\ldots(1-\frac{1}{\frac{(n)(n+1)}{2}})^2 \; \;\forall n \geq 2$

$$a_n = \left(1-\frac13\right)^2\cdot\left(1-\frac16\right)^2\ldots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2 \; \;\forall n \geq 2$$ I have no idea how to solve this, however I will give it a try ...
1
vote
2answers
64 views

Find a Cauchy sequence that doesn't $p$-converge to any rational number.

Let $p$ be a prime number. For any ratinoal number $x$, define $$|x|_p = \begin{cases} 0 \,, & \mbox{if } \,x=0 \\ p^{-\alpha}\,, & \mbox{if }\,x=p^\alpha\frac{n}{m} \,\,,\mbox{in which }m,n\...
-2
votes
1answer
47 views

Prove that if $\sum_{n = 1}^{\infty} a_{n}$ converges, then $\sum_{k = 1}^{\infty} A_{k}$ converges(2) [duplicate]

Suppose that $\{a_n\}$ be a sequence of real numbers.let $\{n_{k}\}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k \in \...
2
votes
1answer
41 views

Showing $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$

Question: Let $f:(0,1) \to \mathbb{R}$ be a differentiable function such that $|f'(x)| \leq 5$, for all $x \in (0,1)$. Show that the sequence $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$. ...
-1
votes
2answers
86 views

Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous [duplicate]

Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous? we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my ...
2
votes
1answer
59 views

If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Let $\{a_n\}$ be a bounded and non-decreasing sequence of reals, and $b_n = n(a_{n+1} - a_{n})$. (a) Show that $\liminf b_n = 0$ (b) Give an example of a sequence $\{a_n\}$ such that $\{b_n\}$ ...
3
votes
2answers
50 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 ...
1
vote
2answers
77 views

find the point of convergence of sequence {$a_n$} [duplicate]

Let $\displaystyle a_n= \sum_{k=1}^{n} \frac{n}{n^2+k}$, for $n\in \mathbb{N}$. Then what is the nature of sequence $\{a_n\}_{n\in\mathbb{N}}$. I tried using the Cauchy's general principle of ...
0
votes
1answer
35 views

Limit of sequence of sequences

I got to thinking about sequences of Cauchy sequences. Here is a simple example. Let us define $b_n = (n,1,1,1,...)$ for $n\in\mathbb N$. So we have \begin{align} b_1&=(1,1,1,1,...)\\ b_2&=(2,...
1
vote
2answers
75 views

For a sequence $(a_n)$ of real numbers, $\sum_{n=1}^\infty |a_{n+1}-a_n|$ converges implies $(a_n)_{n\in\mathbb{N}}$ converges.

I know that when the series $\sum_{n=1}^\infty |a_{n+1}-a_n|$ converges, then we have $|a_{n+1}-a_n|\rightarrow 0$ So by using this I was going to prove that the sequence $a_n$ is Cauchy. But couldn't ...
1
vote
1answer
35 views

Sequences of sequences: question about Cauchy's construction of the real numbers

As is well known, one way of constructing the real numbers is to consider Cauchy sequences and call two of them equivalent if they have the same limit. I got to thinking about the Cauchy sequences ...
0
votes
1answer
52 views

Confusion regarding Cauchy's General Principle and Uniform Convergence

The definitions of the two are so alike, that it confuses me. Cauchy's General Principle: The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x ...
1
vote
2answers
61 views

Prove this sequence converges or diverges to $-\infty$

Let $a_n$ be a sequence such that for every $n$: $a_n\le\frac{1}{2}(a_{n-1}+a_{n-2})$. Prove that $a_n$ either converges to a real number $L$ or diverges to $-\infty$ $(L\in[-\infty,\infty))$. ...
0
votes
1answer
96 views

Comparison Test for complex series: logical argument?

I am trying to show that the comparison test holds for complex series, meaning: if $\sum_{n=0}^{\infty} z_n $ is a complex series and $\sum_{n=0}^{\infty} a_n $ is a convergent non-negative real ...
0
votes
1answer
51 views

Convergence of a sequence defined by a recurrence inequality [duplicate]

Let $(u_n)$ be a sequence of real numbers with the following property \begin{eqnarray*} \forall\,m,n\ge1,\quad u_{m+n}\le \frac{m}{m+n}u_m+\frac{n}{m+n}u_n \end{eqnarray*} Is it true that $(u_n)$ ...
1
vote
2answers
39 views

Meaning of being a Cauchy sequence

A Cauchy sequence in a metric space $(X,d)$ is a sequence for which the distance between two terms can be made as small as we want, provided we look far enough in the sequence. Let $X \subseteq Y$, ...
2
votes
2answers
55 views

Please help me to understand a proof that $a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$ is a Cauchy sequence

$\textbf{Problem:}$ Show that $$a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$$ is a Cauchy sequence. This is the solution. Here's my question: In the first line, we have $...
7
votes
1answer
138 views

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Given a sequence $\{S_n\}$, $n\in\Bbb N$: $$ S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!} $$ Prove that $S_n$ converges for all $x\in\Bbb R$. Please note that i know $S_n$ is a simple ...
3
votes
3answers
78 views

Proof verification. $\{x_n\}$ is a sequence such that $|x_{n+1} - x_n| \le C\alpha^n$ for $\alpha\in (0, 1), n\in\Bbb N$. Prove $x_n$ converges.

Let $\{x_n\}, n\in \Bbb N$ denote a sequence such that: $$ \begin{cases} |x_{n+1} - x_n| \le C\alpha^n \\ 0 < \alpha < 1 \end{cases} $$ Prove $\{x_n\}$ converges. Given the fact $|x_{n+1} - ...
-1
votes
1answer
54 views

Sequence of function on $\mathbb{R}$ Cauchy iff convergent

Theorem: Let $(f_n)$ be a sequence of functions on $I \subseteq \mathbb{R}$. Then $(f_n)$ pointwise convergent iff pointwise cauchy. Here, I only prove "$\Longleftarrow$" since the converse is very ...
2
votes
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 ...
0
votes
1answer
59 views

Prove that $\{x_n\}$ is a Cauchy sequence iff $\forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$

Let $\{x_n\}$ denote a sequence. Prove that: $$ \{x_n\}\ \text{is fundamental} \iff \forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon $$ Let $P$ be a ...
5
votes
1answer
85 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 ...
2
votes
1answer
33 views

Proof verification of $x_n = \sum_{k=1}^n a_kq^k$ is Cauchy given $|a_k| \le C, |q| < 1, k\in\Bbb N$

Given a sequence $\{x_n\}$: $$ x_n = \sum_{k=1}^n a_kq^k $$ and: $$ \begin{cases} |a_k| \le C\\ |q| < 1\\ k\in\Bbb N \end{cases} $$ Prove $\{x_n\}$ is a fundamental sequence. By ...
3
votes
2answers
65 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 ...
1
vote
2answers
34 views

Identical Cauchy sequences and continuity.

Find a set $X$ and two metrics $d$ and $m$ on $X$ such that the Cauchy sequences of $(X,d)$ and $(X,m)$ are identical and the identity map from $(X,d)$ to $(X,m)$ is continuous but not uniformly ...
3
votes
0answers
78 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 ...
0
votes
3answers
64 views

Show that if two metrics induce the same topology, one metric space is compact iff the other one is.

Question: Given two metric spaces $(X, d)$ and $(X,e)$, where $\tau(X,d)$ = $\tau(X,e)$, show that $(X,e)$ is compact if and only if $(X,d)$ is compact. I'm aiming to prove this using the definition ...
3
votes
2answers
88 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 ...
2
votes
2answers
63 views

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

Let $x>0$, $x_1=\cos(x),$ and $x_{n+1}=\cos(x_n), \forall n\geq1.$ Then the sequence $(x_n)_{n\geq 1}$ is bounded but not monotone, bounded but not Cauchy, Cauchy, convergent. It ...
0
votes
1answer
35 views

Prove that $\{x^n\}$ is Cauchy in $S\subseteq \ell_\infty$

I'm kind of new into Functional analysis. So, I have the following question bothering me. Let $S$ be the set of sequences having only a finite number of non-zero terms. Clearly, $S\subseteq \ell_\...
0
votes
1answer
23 views

Show that there exists a measurable set $A\in\mathcal{A}$ such that $\chi_{A_n}\to\chi_{A}$ $\mu$-a.e.

I'm studying for measure theory and I am wondering how to understand a line in the proof, as well as how the problem could be solved without referencing the fact that $L_p$ spaces are Banach: ...
0
votes
1answer
24 views

Prove this is a cauchy sequence [duplicate]

Let ${a_n}$ be a sequence such that there exists an M > 0 such that for all n ∈ N one has $|a_{n+1} − a_n|$ ≤ M/$2^n$ Prove that ${a_n}$ is a Cauchy sequence. My attempt: I tried to use the triangle ...
1
vote
2answers
66 views

Cauchy´s convergence test for Series

Show whith the cauchy's convergence test for series, that the sequence : $$ b_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} $$ converges. I think it would be not so difficult to ...
3
votes
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 ...
1
vote
1answer
44 views

Show that $ \lim_{n\to\infty} \int_{0}^{1} f_n $ exists

Let $(X,d)= (C[0,1],d)$ where $C[0,1]$ is the set of real-valued continuous functions on $[0,1]$ and $d= \int_{0}^{1} |f-g|$ is the Riemann Integral. Suppose $(f_n)$ is a Cauchy sequence in $(X,d) $ ...
0
votes
0answers
19 views

Is there a space where no convergent Cauchy sequences exist?

I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces ...