Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

132 questions
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 $f: (a, b) \rightarrow \mathbb{R}$. ...
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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 ...
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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 ...
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Converge uniformly on open interval implies on closed interval

Suppose $f_n(x)$ is defined on $[a,b]$, and it converges uniformly to $f(x)$ on $(a,b)$. And the sequences $f_n(a)$ and $f_n(b)$ both converge (say, to points $c$ and $d$ respectively). I want to ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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. ...
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