Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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1answer
32 views

Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon $, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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3answers
44 views

Intuitively, $\sqrt{n}$ is not convergent. However, $|\sqrt{n+p}-\sqrt{n}|<\epsilon, \forall \epsilon>0, p\geq 1$

$|\sqrt{n+p}-\sqrt{n}|<\epsilon$ Clearly, $|\sqrt{n+p}-\sqrt{n}|=\frac{p}{\sqrt{n+p}+\sqrt{n}} \leq p/\sqrt{n} \rightarrow 0$. But by definition of a cauchy sequence, if we can choose $\exists N: ...
0
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1answer
36 views

Determine if Sequence is Cauchy

Can someone please tell me how to determine if a sequence is Cauchy without using the limit. I'm supposed to use partial fraction decomposition $a_n=\frac{1}{n(n+1)}$. When I did the partial ...
0
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1answer
27 views

The convergence of a bounded sequence ${x_n}$ satisfying $x_{n+1} - \epsilon_n \le x_n$, where $\sum_{n=1}^\infty \epsilon_n$ is absolutely convergent

Statement: If a bounded sequence $\{x_n\}_{n=0}^\infty$ in $\mathbb{R}$ satisfies $x_{n+1} - \epsilon_n \le x_n$ for $n \in \mathbb{N}$, where $\sum_{n=1}^\infty \epsilon_n$ is an absolute convergent ...
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2answers
44 views

Cauchy sequence in a discrete space

how to prove that any Cauchy sequence in a discrete space is stationary Let $(x_n)$ be a cauchy sequence then $$\forall \varepsilon>0, \exists n_0\in \mathbb{N},\forall p,q \geq n_0\Rightarrow \...
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0answers
18 views

Notation in Cauchy Sequence and Convergence

I have been reading lecture script and wasn't sure where this inequation for a fix-point iteration comes from and what it means. For $k, j \geq 0$ $$ \left|x_{k+j}-x_{k}\right| \leq \sum_{i=0}^{j-1}\...
0
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1answer
33 views

Proving that $l_\infty$ is complete

I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question - And the books' answer: What I don't understand in the proof - Why can we understand ...
5
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1answer
67 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 ...
0
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2answers
47 views

$n$ tending to infinity

What does n tends to $\mathbf\infty$ mean ? Is it equivalent to saying $n>K$, $\;K\in \Bbb N$ ? For a Cauchy sequence, is $|a_m-a_n|<\varepsilon, \enspace m,n >K$ equivalent to saying $a_m-...
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0answers
35 views

Direct proof for bounded monotonic sequence is Cauchy

Show that every bounded and monotonic sequence is Cauchy. My proof: Suppose $(a_n)$ is a bounded and monotonic sequence. Since $(a_n)$ is bounded, it follows that $\exists M \in \mathbb{Q}^{+}$ ...
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1answer
51 views

Convergence of $\sum_{j=1}^\infty x_j^2$ assuming that the sum of squares is finite.

If I let $l_2$ be the set of all real sequences $\{x_j\}_{j\in N}$, such that $\sum_{j=1}^{\infty} x_j^2 < \infty$, is there any way to show that this sum converges? Can I do it by showing that $\...
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3answers
50 views

Is the following sequence $x_{n}=\frac{1 +(-1)^n}{n}$ Cauchy?

Is the following sequence $x_{n}=\frac{1 +(-1)^n}{n}$ Cauchy? I got not Cauchy, but would appreciate someone to check this. Thank you! So I got the sequence is bounded by 2 and that $abs(x_n-x_{n+1})&...
0
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1answer
28 views

Proving a sequence is Cauchy from uniform continuity

Given: $f: X \longrightarrow Y$ is uniformly continuous on $X$, $(x_n)_n \in X $ is a Cauchy sequence. Question: What can you say about the sequence ${f(x_n)}$ ? My attempt: Since $f$ is uniformly ...
11
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4answers
2k views

Can a Cauchy sequence converge for one metric while not converging for another?

Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only ...
0
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2answers
46 views

Some limits of sequences

Prove by definition the following limit: $$\lim_{n\rightarrow \infty}\sqrt{\frac{2n^3+3n-1}{8n^3+n^2}}=\frac{1}{2}$$ For the first i've been tried rationalize: $$\begin{align}\left|\frac{\sqrt{2n^3+...
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1answer
57 views

Proving a sequence is Cauchy in metric

Consider the sequence, $f_n(x)= \begin{cases} (2x)^n & 0 \leq x\leq \frac{1}{2} \\ 1 & \frac{1}{2} \leq x \leq 1\\ \end{cases}$ Then we need to show that $\{ f_n\}$ is Cauchy ...
0
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2answers
44 views

Prove that if $n\ge 2$ is an integer, then $1/n^2 < 1/(n-1) - 1/n$ [closed]

Prove that if $n\ge 2$ is an integer, then $1/n^2 < 1/(n-1) - 1/n$ Unsure how to start this question! I've tried to prove by induction but I keep getting $n>1$?
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2answers
44 views

How these two definitions are equivalent?

Please have a look at these two equivalent defintions of Cauchy's general principle of convergence series. I understand the first defintion but I'm having problems with the second defintion. First ...
3
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3answers
42 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 ...
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1answer
41 views

Suppose $f:[0,1]\times[0,1]\mapsto X$ is a continuous function. Show that $[0,1]\times[0,1]$ can partitioned into rectangles s.t $f(R_i)\subseteq U_k$

Suppose $f:[0,1]\times[0,1]\mapsto X$, is a continuous function where $X$ compact and connected subset of $\mathbb{R}^n$. Show that $[0,1]\times[0,1]$ can partitioned into rectangles $R_i$ such that $...
2
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1answer
17 views

Completeness of a Normed Space of Smooth, Bounded Functions

As part of a proof of the Picard–Lindelöf theorem, I am using the following space: $X = \{ u \in C([0,T]) : u(0) = \alpha , || u - \alpha || \leq K\}$ where $K \in \mathbb{R}_{> 0} , \ \alpha \in ...
0
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1answer
52 views

Series limit involving Binomial coefficients

Consider the parameters $v_{a}$, $v_{b}$ be such that $0<v_{a}\leq v_{b}$ and $c>0$. I have an equation involving the Binomial distribution that I need to solve with respect to $p(n)$: $\sum_{k=...
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0answers
28 views

Lower bound $\sum\limits_{n=1}^\infty a_n/(a_n -1)$ in terms of $a_1$, where $a_n$ is an infinite sequence under some conditions.

This is a question related to a previous question ($a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. Prove that $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ ...
0
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1answer
28 views

$(f(x))_{n\in\mathbb{N}}$ and $(f(y))_{n\in\mathbb{N}}$ have the same limit.

Assume that $f: \mathbb{R} -\{0\}\to \mathbb{R}$ is uniformly continuous. Assume $(x_n)_{n\in\mathbb{N}}\in(\mathbb{R}-\{0\})^\mathbb{N}$ and $(y_n)_{n\in\mathbb{N}}\in(\mathbb{R}-\{0\})^\mathbb{N}$ ...
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2answers
50 views

Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
0
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1answer
63 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
1
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2answers
33 views

If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that : $$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$ Show that $A$ is invertible. Attempt/...
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3answers
72 views

Show $\{x_n = \sqrt{n}\}$ is not Cauchy sequence

Consider the sequence {$x_n$},$x_n$=$\sqrt{n}$ Show that $\forall \varepsilon > 0, \exists n_0 \in \Bbb N$ s.t. $\forall n \geq n_0$, |$x_{n+1}-x_n$|<$\varepsilon$. This is what I have: Let $\...
0
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1answer
41 views

Let $x_n$= $\sum_{k=1}^{n}$ 1/k! = 1+1/2!+1/3!+…+1/n! show that the sequence {$x_n$} is Cauchy.

Let $x_n = \sum_{k=1}^{n} 1/k! = 1+1/2!+1/3!+...+1/n!$ for each $n \geq 0$. Show that the sequence $\left(x_n\right)$ is Cauchy. This is what I have: for $n>m$, \begin{align} &|x_n-x_m| \\ &...
1
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1answer
44 views

Proving that the metric space of all sequences of positive integers is complete

Consider the set of all sequences of positive integers with the following metric:given $x=(n_j)$, $y=(m_j)$ $$d(x,y)= 1/\inf\{j: n_j \ne m_j\}$$ if $x\ne y$ and $0$ otherwise. I want to show that it ...
1
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2answers
32 views

Complete metric space on unique metric

Consider the metric $$d(m,n) = \frac{|m-n|}{mn}$$ Is this metric on the natural numbers $(1,2,\ldots)$ complete? I'm struggling but heres an idea I have from reading other similar questions. The ...
2
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3answers
62 views

Proving that a sequence converges if $|a_n - a_{n+1}| < Mr^n$ for some $M > 0$ and $r \in (0,1).$

Let $\{a_n\}_n$ be a sequence. Suppose that there exist $M \gt 0$ and $r \in (0, 1)$ such that $|a_n - a_{n+1}| \lt Mr^n$ for all $ n \in \Bbb N.$ Prove that $\{a_n\}_n$ converges. I'm not really sure ...
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2answers
20 views

Verification of proof of nonconvergence

I'm not sure if my proof is sound or not, I was wondering if anyone could verify. Prove that the sequence $\{a_n\}_n $ given by $a_n = \frac{(-1)^n\;n+1}{3n}$ does not converge. Proof: Let $\epsilon ...
0
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1answer
26 views

Let there be a sequence such that the distance between two consecutive terms converges to 0. Must this sequence converge? [duplicate]

I'm trying to solve the following analysis problem and I've developed a proof, I'm just not entirely sure if it's valid or not. Let $\{a_n\}_n$ be a sequence such that $|a_n -a_{n+1}|\to 0$. Must $\{...
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 $(...
1
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1answer
59 views

Does $\lim\limits_{x \rightarrow c} f(x)$ exist if the sequence $\{ f(x_n)\}_{n=1}^\infty$ is Cauchy?

I'm struggling a little with this question: Let c be a cluster point of $A ⊂ \mathbb{R}$, and $f : A → \mathbb{R}$ be a function. Suppose for every sequence $\{x_n \}$ in A, such that $\lim x_n = c$,...
2
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1answer
37 views

False theorem on particular case of Cauchy Sequences — why is it wrong?

It goes as follows: If $|x_{n+1} - x_n| < \epsilon$, $\{x_n\}$ is Cauchy. We have: $$\forall \epsilon_0 > 0, \exists N : n > N \implies |x_{n+1} - x_n| < \epsilon_0$$ Let $m,n \in \mathbb{...
2
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1answer
39 views

Question on operator norm $\vert \vert \cdot \vert \vert$

I have just been introduced to the operator norm $\vert \vert \cdot \vert \vert$ on $L(X,Y)$ where $X, Y$ are respective normed spaces and any $T \in L(X,Y)$ is simply a linear map from $X$ to $Y$. ...
0
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0answers
49 views

Proof of divergence for complex number non-cauchy sequence

So, I trying to prove the following: "Show that a complex number sequence converges, iff, it is a Cauchy sequence". Let $\{w_n\}$ be a complex number sequence. For every $\epsilon>0$,$\exists N=N(\...
1
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1answer
35 views

Is this normed vector space complete?

The normed vector space in question is $(C^{1}[0,1],\lvert \lvert \cdot \rvert \rvert _c )$ where $\lvert \lvert f \rvert \rvert _c := \lvert f(1) \rvert + \lvert \lvert f' \rvert \rvert _{1}$ For ...
0
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1answer
43 views

Suppose V is a linear subspace of $\mathbb{R}^n$. How to show it's closed in $\mathbb{R}^n$

We want to show that if $V\subseteq\mathbb{R}^n$ is a linear subspace of $\mathbb{R}^n$, then it's closed. Let $V\subseteq\mathbb{R}^n$ arbitrary e suppose $V$ linear subspace of $\mathbb{R}^n$. As ...
0
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1answer
45 views

Multiplying convergent sequence by divergent sequences of natural numbers

Let $\{x_k\}$ be a sequence converging to the limit $a \ne 0$ and let $\{n_k\} \subset \mathbb{N} $ be a divergent sequence of natural numbers. I want to show that the sequence $\{x_k n_k\}$ is ...
2
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1answer
31 views

If we have a sequence of measurable functions that is Cauchy with respect to the weak L^p norm, is it Cauchy with respect to the L^p norm?

If $(f_n)$ measurable on $(X,\mathcal{M},\mu)$, $f_n$ Cauchy with respect to weak quasi $L^p$-norm: $[f_n]_p=\sup_{\alpha>0}\alpha \lambda_{f_n}(\alpha)^{\frac{1}{p}} $ where $\lambda_{f}(\alpha)...
0
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1answer
28 views

Is the sequence converging?

Consider the sequence x1, x2, x3 ... defined by Xn={3+(1/n)} (if n is odd) and {4+(1/n)} (if n is even) As n→∞, the value of the first sequence will be 3 and the second one will be 4. Does this mean ...
0
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1answer
44 views

Cauchy sequence and boundedness

We know that every Cauchy sequence is bounded. and the converse may not be true. but if we define a sequence $x_n=n$ with metric $d(m,n)$=$\lvert \dfrac{1}{m}-\dfrac{1}{n}\rvert$. then this is a ...
0
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2answers
30 views

Limit evaluation using algebra of sequences and sequence theorems

By making use of only the theorems on sequences (ex: algebra of sequences/cauchy's first theorem of sequences/limit of geometric mean of a sequence etc), how to prove the following: $lim_{n\to\infty}(...
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2answers
75 views

Proof that the metric space $M$ is complete if every closed ball of $M$ is complete.

Let $M$ be a metric space I'm asked to prove the statement "Every closed ball of $M$ is complete $\implies$ $M$ is complete". My attempt at this is as follows: Let $\{y_i\}$ be a cauchy ...
2
votes
2answers
78 views

Show that sequence is a Cauchy sequence

Prove that given sequence $$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence My attempt : $|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\...
1
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2answers
72 views

Proving that $Y$ is complete

Let $X$ be a dense subset of $Y$. Let every cauchy sequence in $X$ converge to a point in $Y$ $(1)$. By the definition of dense I know that every point in $Y$ is either in $X$ or is a limit point ...
0
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0answers
45 views

Prove that a Cauchy sequence in $\mathbb{R}^n$ is convergent. [duplicate]

I need some help with this problem, I've seen that there is a similar problem about proving that Euclidean Space is a complete metric space, but I haven't learn about Euclidean or metric spaces and ...