# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

143 questions
Filter by
Sorted by
Tagged with
11k views

### Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
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 ...
2k views

32k views

### Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy

Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy. My Work So by the definition of a Cauchy sequence, for ...
4k views

### Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be ...
3k views

### Pseudo-Cauchy sequence

I have never seen this terminology before, so I will provide the given definition. A Pseudo-Cauchy sequence is : A sequence $(a_n)$ if for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such ...
679 views

### Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't infinite. ...
5k views

### Metric space is totally bounded iff every sequence has Cauchy subsequence

Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. I think I proved the Cauchy subsequence part: Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be ...
1k views

### If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
6k views

694 views

### Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \ge 0$. Proof: I will attempt to show that the ...
21k views

### Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
12k views

### How do I prove a uniformly continuous function preserves Cauchy sequences?

Let $f$ be a uniformly continuous function on A of $\Bbb{R}$. How do I show that if $a_n$ is Cauchy, then $f(a_n)$ is Cauchy. This is what I have worked on, but it does not quite make sense since I ...
3k views

### Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$

Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$. Let $f: (a, b) \rightarrow \mathbb{R}$. ...
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 ...
4k views

### Show that every monotonic increasing and bounded sequence is Cauchy.

The title is kind of misleading because the task actually to show Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy without knowing that: Every bounded non-empty ...
4k views

### Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$

Past final exam question for an intro to Real Analysis course: Let $C > 0$, $0<r<1$ and suppose that $\forall n\in \mathbb N, |x_{n+1} - x_n| \leq Cr^n$. Please help me prove that $(x_n)$ is ...
1k views

3k views

### Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences ...
681 views

### A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
982 views

### Show that a Cauchy sequence has a fast-Cauchy subsequence

A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})<+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence. **My attempt:**Argue by contradiction, ...
149 views

### I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy [duplicate]

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy ? I tried to find such sequence $x(n)=1/2,1/3,1/2,1/3,1/4,1/2,1/3,1/4,1/5,,,,$ it's not Cauchy since it is ...
599 views

### How to prove the sequence $x_{n+1} = \frac{x_n}2 + \frac 1{x_n}$ is a Cauchy sequence

Here is a question I do not know how to prove. Thanks for your helping！ Prove that the sequence $$x_{n+1} = \dfrac{x_n}2 + \dfrac 1{x_n}, x_0 = 1$$is a Cauchy sequence.
386 views

Let $a\leq b \in \mathbb{R}$. Show that the sequence $a_1 = a, a_2=b$ and $a_{n+2}=\frac{a_{n+1}+a_n}{2}$ for $n\geq 1$ is Cauchy and find it's limit. I did for $n>m$: $$|a_n-a_m|=\frac{1}{2}|(a_{... 3answers 221 views ### Prove that the sequence a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right) is convergent and find its limit Let c>0, a_{1} = 1, and$$a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$$I need to: Show that a_{n} is defined for every n\geq 1 Show that this sequence is convergent. Find its ... 1answer 4k views ### Continuous Functions and Cauchy Sequences We know that if a function f: A \mapsto \mathbb{R}, A \subseteq \mathbb{R}, is uniformly continuous on A then, if (x_n) is a Cauchy sequence in A, then (f(x_n)) is also a Cauchy sequence. ... 1answer 469 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 ... 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 ... 2answers 1k views ### Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces. Let  X  and  Y  be metric spaces, and let  f: X \to Y  be a mapping. Determine which of the following statements is/are true. a. If  f  is uniformly continuous, then the image of every Cauchy ... 3answers 174 views ### Construct such d that (\mathbb{R} \setminus \mathbb{Z}, d) is complete metric space Good evening! I had topology exam yesterday and I had a question that gave me problems. Lets consider \mathrm{G} = \mathbb{R} \setminus \mathbb{Z}  , i.e. real line without integrals, where \rho ... 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 ... 1answer 6k views ### Proving that product of two Cauchy sequences is Cauchy Given that x_n and y_n are Cauchy sequences in \mathbb{R} , prove that x_n y_n is Cauchy without the use of the Cauchy theorem stating that Cauchy \Rightarrow convergence. Attempt: Without ... 3answers 957 views ### Prove that a_n=1+\frac{1}{1!} + \frac{1}{2!} +…+ \frac{1}{n!} converges using the Cauchy criterion Any tips on how to approach these kind of proof problems when a factorial is included? Here is what I've tried, By the Cauchy criterion the sequence converges if for every \varepsilon>0 there ... 5answers 640 views ### Prove the following sequence is a Cauchy Sequence Let \{X_n\} be the sequence defined recursively by x_1=2 and x_{(n+1)}=(x_n/2)+(5/x_n). Prove that \{x_n\} converges and find the limit of the sequence. I understand the definition of a ... 1answer 192 views ### lemma: Cauchy sequences are bounded. Lemma: Cauchy sequences are bounded. Proof: Given Cauchy sequence (s_n), for \varepsilon=1 we obtain N\in\Bbb N such that m, n > N implies |s_n − s_m| < 1. (2) implies |s_n − s_{N+1}... 1answer 3k views ### The space of continuous functions C([0,1]) is not complete in the L^2 norm I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in C([... 0answers 34 views ### Prove that the system of real numbers is complete by Cauchy sequence 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 ... 1answer 573 views ### Proof that the series expansion for exp(1) is a Cauchy sequence Consider the series expansion for the exponential function at x = 1:$$a_n := \sum\limits_{i = 0}^n{\frac{1}{i!}} I want to prove that this is a Cauchy sequence, using the remainder formula for ...
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 ...
I have been tasked with showing that for a metric space $(X,d)$, a subset $E \subseteq X$ is relatively compact $\iff$ $E$ is totally bounded. I believe I have shown the forward implication \$(\...