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.

6
votes
0answers
543 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
votes
0answers
600 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
votes
0answers
198 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
votes
0answers
162 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
votes
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
votes
0answers
147 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 $\...
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 ...
3
votes
0answers
57 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
votes
0answers
276 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 ...
3
votes
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
votes
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
votes
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
votes
0answers
297 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
votes
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
votes
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
votes
0answers
29 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
votes
0answers
47 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
votes
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
votes
0answers
49 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
votes
0answers
55 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
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 ...
2
votes
0answers
69 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
votes
0answers
74 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
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
votes
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
votes
0answers
146 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
0answers
263 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
0answers
133 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
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
153 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
558 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
0answers
151 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 ...
2
votes
0answers
229 views

Prove that the sequence $\{x_n \}$ is Cauchy

Let $\{x_n\}$ be a sequence of real numbers with $|x_n − x_m| < 1/(n+m)$ where $n,m$ are positive integers. . k  Prove that the sequence $\{x_n \}$ is Cauchy. Here is what I have so far.. Let $\...
2
votes
0answers
107 views

Subspace of a weakly sequentially complete is weakly sequentially complete

A Banach space $X$ is called weakly sequentially complete if all weakly Cauchy sequences are weakly convergent. Question: If $Y$ is a subspace of a Banach space $X$, must $Y$ be weakly sequentially ...
2
votes
0answers
97 views

Is my proof of uniform convergence valid, using summation by parts?

Edit: I revised the choice of $N=max(N_1,N_2)$, in order to give slightly sharper estimates to make sure I can say "less than epsilon" at the end, and not be stuck with a 3M$\epsilon$. Is my proof ok?...
2
votes
0answers
74 views

Find a metric on $\mathbb{R}$ with the property that the sequence of natural numbers is Cauchy.

I'm trying to solve the following question: Find a metric $d$ on $\mathbb{R}$ that is equivalent to the usual metric and has the property that the sequence $(n)_{n=1}^{\infty}$ is a Cauchy sequence. ...
2
votes
0answers
132 views

Cauchy sequence in reproducing kernel Hilbert space

Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} \...
2
votes
0answers
27 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] \...
2
votes
0answers
55 views

cauchy sequence integers

let $\{p_n\}$ be a sequence of integers. prove that if $\{p_n\}$ is cauchy, then $\{p_n|n \in N\}$ is finite. my proof: don't know what else do it need or is it right please help... pf: $Given, \{p_n\...
2
votes
0answers
322 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ x\...
2
votes
0answers
157 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, $$...
2
votes
0answers
55 views

Show, using only the definition, that if $\{X_n\}$ is Cauchy and $C\in\mathbb{R}$ then $\{CX_n\}$ is Cauchy.

Let me know if what I did this correct please. Let $\epsilon>0$ be given, we want to find $N\in\mathbb{N}$ such that $|CX_n-CX_m |<\epsilon$ $\forall n,m\geq N$. $$|CX_n-CX_m |=|C||X_n-X_m |$$ ...
2
votes
0answers
2k views

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{ 2^2}+\cdots+\frac{1}{n^2}$

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{2^2}+\frac{1}{3^2}+ \ldots +\frac{1}{n^2}$ My attempt Take $|x_m-x_n|$, where $m>n$, We have $|x_m-x_n|=\frac{1}{(n+1)^...
2
votes
0answers
273 views

Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by \begin{align}...
1
vote
0answers
23 views

Why is it necessary to first reduce our case to a finite $J_{2} \in \mathbb N$ to show completeness of $\ell^{\infty}$

Let $(x^{(n)})_{n}\subset\ell^{\infty}$ be a Cauchy sequence (w.r.t. $\vert \vert \cdot \vert \vert _{\infty}$). Thus for any $\epsilon >0 $ there exists $N \in \mathbb N$ so that for all $n,m \geq ...
1
vote
0answers
68 views

Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a Cauchy sequence.

Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a cauchy sequence. I'm just making sure my logic and reasoning is sound for the above proof: Definition of cauchy sequence: $f_n$ is Cauchy if for all ...
1
vote
0answers
23 views

Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
1
vote
0answers
54 views

Prove that $\Bbb Q$ is dense in $\Bbb R$ constructed by Cauchy sequences

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...