Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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6
votes
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 ...
2
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2answers
108 views

Proving Cauchy Given Given Sequence Terms Arbitrarily Close

I need to show that $\{x_{n}\}$ is Cauchy given that there exists $0<C<1$ s.t. $|x_{n+1}-x_{n}|\leq C|x_{n}-x_{n-1}|$. Intuitively, that statement clearly implies $\{x_{n}\}$ is Cauchy, since ...
2
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4answers
118 views

If $(a_{2n+1})$ and $(a_{2n})$ converge to $a$ then $(a_n)$ converges to $a$.

If $(a_{2n+1})$ and $(a_{2n})$ converge to $a$ then $(a_n)$ converges to $a$. So far I realize that if $(a_{2n+1})$ and $(a_{2n})$ converge then for each $\epsilon>0$, there exists $N$ such that ...
0
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2answers
86 views

Question on Cauchy Sequence definition?

Kaplansky defines a Cauchy sequence if for any $\epsilon > 0$ there exists sufficiently large $i, j$ such that $D(x_i, x_j) < \epsilon$ for some sequence $\{x_n\}$ in a metric space. The ...
2
votes
0answers
276 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
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0answers
176 views

On Cauchy Sequences

I would consider this a soft question because I am seeking some insight on how to work with Cauchy sequences by using the Cauchy criterion for convergence. To my understanding, the definition is ...
2
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1answer
1k views

An example of a bounded pseudo Cauchy sequence that diverges? [duplicate]

Harmonic series diverges and pseudo Cauchy however it's not bounded. So how can I find such a sequence? A sequence $(s_n)$ is pseudo-Cauchy if, for all $\xi>0$, there exists an $N$ such that if $n ...
10
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2answers
807 views

Proving completeness of Nikodym Metric

I'm trying to prove completeness directly of the metric given by $d(A, B) = \mu (A \triangle B)$ on a finite measure space $(X, M, \mu)$. Edit: I should make clear that I'm referring to completeness ...
2
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2answers
273 views

Cauchy sequences and condition $X_{n+1} – X_n\to 0$

Let $\{X_n\}$ be a sequence and suppose that the sequence $\{X_{n+1} – X_n\}$ converges to $0$. Give an example to show that the sequence $\{X_n\}$ may not converge. Hence, the condition that $|X_n-...
2
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1answer
103 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: $$\sum_{i=1}^...
1
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1answer
313 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
3
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1answer
410 views

Show that $ x_n = \left(1 + \frac{r}{n} \right)^n $ has an upper bound

I asked this question but maybe my doubt was not enough clear. So I will ask something more specific: Show the sequence $x_n = \left(1 + \frac{r}{n}\right)^n$ for $ r \in \mathbb{Q}, r>0$ has an ...
1
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3answers
3k views

Limit of sequences: $ \lim (1 + \frac{r}{n} )^n = e^r $

Consider the sequence $x_n = \left (1+\frac{r}{n} \right )^n $ for $r \in \mathbb{Q} $. I need to prove that $ \lim x_n = e^r $ My attempt of proof (for r>0) is to find a subsequence of $x_n$ that ...
0
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1answer
3k views

Is this valid: Every Cauchy sequence in a normed space is absolutely convergent.

Proof. Let $X$ be a normed space with norm $|\cdot |$ and $(x_n)$ be Cauchy. Then for all $\epsilon \gt 0, \ \exists N : m,n \gt N \implies |x_m - x_n| \lt \epsilon$ is the standard definition of ...
3
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2answers
257 views

Determining if a sequence of functions is a Cauchy sequence?

Show that the space $C([a,b])$ equipped with the $L^1$-norm $||\cdot||_1$ defined by $$ ||f||_1 = \int_a^b|f(x)|dx ,$$ is incomplete. I was given a counter example to disprove the statement: Let $...
0
votes
1answer
36 views

Cauchy sequence in valued fields

I can't understand this property, left unproved by my textbook as a trivial fact: let $K$ be a valued field, with valuation $\left|\phantom{x}\right|:K\longrightarrow\mathbb{R}$, let $\{a_n\}$ be a ...
2
votes
3answers
6k views

How to prove $(-1)^n$ is not Cauchy in $\mathbb{R}$?

I am an engineer and I would like to prove that the sequence $(-1)^n$ is not a Cauchy sequence, in order to understand the definition of Cauchy sequence better. Thank you.
6
votes
1answer
174 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. $$a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$$
2
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1answer
97 views

On constant sequences

I am a non-mathematician who is taking a self-help course on real analysis. I want to prove that a constant sequence $\{x_n\}_{n=1}^{\infty}$ is Cauchy. I know it is true since $|x_n-x_m|=0$ for all $...
0
votes
1answer
476 views

Proving a Sequence is Cauchy

Suppose a sequence $\{a_n\}$ has this property: there exist constants $C$ and $K$, with $0<K<1$ such that $\vert a_n - a_{n+1} \vert < CK^n$, for $ n \gg 1$. Prove that $\{a_n\}$ is a Cauchy ...
0
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1answer
272 views

Cauchy Sequences and Integrals

Suppose $f(x)$ is continuous and decreasing on $[0,\infty]$, and $f(n) \to 0$. Define $\{a_n\}$ by $a_n = f(0)+f(1)+ \cdots + f(n-1) - \int_{0}^{n} f(x)dx$. a). Prove $\{a_n\}$ is a Cauchy sequence ...
0
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1answer
823 views

Proof for Term by Term Differentiation

Show that the series for which the sum of first n terms $f_n(x)=\frac{nx}{1+n^2x^2}, 0\leq x\leq 1$ can not be differentiated term by term at $x=0$. What happens at $x\neq0$. I have found similar ...
26
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2answers
25k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
1
vote
1answer
573 views

Proof of the completion of a metric space using cantors diagonal argument and showing a diagonal sequence is cauchy

I am studying applied functional analysis out of Applied Analysis by John Hunter. In chp. 1 of the text it gives a proof for the completion of a metric space. I am having trouble with understanding ...
2
votes
1answer
136 views

A question on Cauchy sub-sequences in a metric space $(X,d)$

Let $(X,d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of positive $\epsilon_k \rightarrow 0$, there is a ...
2
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1answer
3k views

Equivalent Cauchy sequences.

Hi everyone I'm having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I'm wondering if the next reasoning is correct or maybe needs some ...
1
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2answers
592 views

Understanding why $L^{2}(\mathbb{R}^{n})$is incomplete

So I have just started a course which threw us right into the space that contains all continuous real valued functions. In other words, for $1 \leq p < \infty$, $$L^p(\mathbb{R}^n) = \{ f : \mathbb{...
8
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3answers
20k views

How to generate a Cauchy random variable

How do I calculate a Cauchy random variable and how do I calculate the probability mass function to show it is "heavy tailed"
1
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2answers
325 views

Cauchy complete subspace of a metric space is closed.

Is some version of the Axiom of Choice required to show that a complete subspace $A$ of a metric space $X$ is closed? By closed I mean that the set's complement is open, or equivalently that it ...
4
votes
3answers
569 views

How can you prove $\{x_n\}$ is a Cauchy sequence if $|x_{n+1} – x_n| \le r|x_n - x_{n-1}|$ for some $0<r<1$?

Let $\{x_n\}$ be a sequence and let $r$ be a number such that $0 < r < 1$. Suppose that $$|x_{n+1} – x_n| \le r|x_n - x_{n-1}|$$ for all $n>1$. Prove that $\{x_n\}$ is a Cauchy sequence. I ...
5
votes
4answers
210 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 ...
23
votes
2answers
6k views

A “non-trivial” example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it ...
5
votes
2answers
402 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. ...
3
votes
2answers
1k views

Convergent Sequence and Cauchy Criterion- Counter Example

Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition: $$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}} \ \ \ for\ all\ n=1,2,3,...$$ Part (1): Prove that the sequence $\...
3
votes
3answers
10k views

Confused with proof that all Cauchy sequences of real numbers converge.

First the textbook proves that all Cauchy sequences are bounded, and so have a convergent subsequence, $\{a_{n_{k}}\}$ that converges to a limit, say $L$. Now we use this to prove that all Cauchy ...
2
votes
1answer
430 views

Uniform Continuity and Cauchy Sequences

Let $(S,d)$ and $(S^*, d^*)$ be metric spaces. If $f:S \to S^*$ is uniformly continuous and if $(s_n)$ is a Cauchy sequence in $S$, then $(f(s_n))$ is a Cauchy sequence in $S^*$. $f:S \to S^*$ is ...
2
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2answers
83 views

On changing from '<' to '$\le$' when taking limits (with norm $|\bullet|_p$)

I'm reading Gouvêa's book on $p-$adic, and there's one problem that I don't think I really get it. Here's a proposition, and the problem attached to it. It's on page 57, 58 of the book. Proposition ...
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 ...
-3
votes
1answer
459 views

Metric Space & Cauchy Sequence

Question: Consider a metric space $(X,d)$ 1) Show that if $x$ is a convergent sequence in $(X,d)$, then it is a Cauchy sequence in $(X,d)$. Suppose $(X,d)$ is complete. Let $f\colon X \to X$ be a ...
1
vote
2answers
58 views

Prove that $\lim \limits_{n\to+\infty}\|\sup_{k>n}u_k-u_n\|=0$

Let $\{u_n\}$ be a Cauchy sequence in the space $(\mathbb R,d)$ with $d(x,y)=\|x-y\|$. Prove that $$\lim_{n\rightarrow+\infty}\|\sup_{k>n}u_k-u_n\|=0.$$ This seems to be obviously however I can not ...
1
vote
2answers
62 views

A specific proof of inequality needed for an analysis problem.

I am working on proving that there exists a Cauchy sequence approaching to $x$ where all the terms of $\{x_j\}$ is strictly less than $x$. My plan is to construct a new sequence as follows. Take ...
1
vote
2answers
162 views

Commutativity of reals using Cauchy seq.

I am trying to prove that real numbers are commutative using the definition of real numbers as the equivalence class of Cauchy sequences of rational numbers. The following is what I know and what I ...
3
votes
2answers
426 views

Uncountably many equivalent Cauchy sequence?

RTP There exists uncountably many Cauchy sequence of rationals that are equivalent. I am trying to solve the above question, and my understanding is that $\Bbb R$ is a set of equivalent classes of ...
0
votes
1answer
655 views

Merten's theorem on cauchy products

Suppose we know that $$\{a_n\}$$ is a sequence such that $$\sum_{n=0}^{\infty}a_n=0$$ and that for some $N$ quite large, we have that $a_n=0$ for any $n\geq N$. Then we notice that the infinite ...
2
votes
1answer
3k views

How to show that space is complete?

Let $N_\alpha=\{(x_n)_{n=1}^\infty\mid \sum_{j=1}^n |x_j|\leq Mn^\alpha\}$, where $\alpha\in R$. Show that $N_\alpha$ is Banach space with the norm $\|(x_n)_{n=1}^\infty\|=\sup_{n\in N} n^{-\alpha} \...
3
votes
2answers
341 views

An issue in constructing the real numbers using Cauchy sequences in the rationals.

I'm having trouble justifying why we can discuss Cauchy sequences before the real numbers are constructed. As we know, the definition of a Cauchy sequence (in the metric space $\mathbb{Q}$) starts ...
7
votes
1answer
1k views

Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges. [duplicate]

Intuitively speaking, I first thought that if the series $\Sigma a_n$ is divergent then $$\lim_{n \to \infty} a_n \ne 0$$ therefore it was clear that $\Sigma \frac{a_n}{1+a_n} $ would be divergent, ...
1
vote
1answer
248 views

Given $\Sigma a_n \to \alpha$ show that $\Sigma \frac{\sqrt{a_n}}{n} \to \beta$ [duplicate]

I am trying to prove that if $\Sigma a_n$ is convergent, then $\Sigma {\sqrt {a_n} \over n}$ is also convergent. I tried to use the comparison test but $\sqrt{a_n} >a_n $ so I couldn't go that ...
5
votes
1answer
589 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(...
0
votes
1answer
579 views

Cauchy Criterion for Sequences as opposed to Series

So through as I've been venturing through baby Rudin I came upon his definition of a cauchy sequence: A sequence $\{ p_n \}$ in some metric space $X$ is said to be cauchy if $$ \forall \; \epsilon &...