# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### 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 (...
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### 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 ...
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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 the set of integers Cauchy complete?

http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
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### Why we define the completeness of a space by the converge of a Cauchy sequence rather than a normal sequence?

The intuition of the completeness to me is that the limit of any sequence converges to the point inside the set itself. But why we define a set to be complete as any Cauchy sequence converge into the ...
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### To prove a sequence is Cauchy [duplicate]

I have a sequence: $a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } }$ , it repeats $n$-times. and i have to prove that it is a Cauchy's sequence. So i did this: As one theorem says that every ...
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### The sequence $x_{n+1}=ax_{n}+b$ converges to where?

$$a,b \in \mathbb R , \ 0\lt a\lt 1 .$$ Define the sequence $$x_{n+1}=ax_{n}+b \text{ for } n\ge0\ .$$ Then for a given $\ \ x_0\ \$ , does this sequence converge? And if it does, to ...
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### Determine if this specific sequence is a Cauchy sequence

I have the following sequence: $$a_n =\sum_{k = 1}^n (-1)^{b_k} {1\over k^2}$$ And the hint is that I have to prove that: $${1\over k^2} < {1\over k-1} - {1\over k}$$ So assuming $m>n$, I ...
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### My Proof: Every convergent sequence is a Cauchy sequence.

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 $\varepsilon \in \mathbb{R}$ ...
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### Existence of $a_k$ such that $\sum_k a_kb_k<\infty$ and $\sum_k a_k=\infty$ given $b_k\to 0$

I was working with a problem from functional analysis. I reduced the problem to the following problem: Let $b_k>0$ be a decreasing sequence converging to $0$. Does there exist a non-negative ...
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### Does $\sum_k \frac{a_k-a_{k-1}}{a_k^2}$ converge if $a_k \uparrow \infty$?

Let $a_n$ be a sequence of strictly increasing postive numbers such that $a_n \uparrow \infty$. Does $$\sum_{k=1}^\infty \frac{a_k-a_{k-1}}{a_k^2}$$ converge? My guess is that it converges. I tried ...
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### If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
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### 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 ...
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### Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
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### If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$

If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$ I feel like answer should be NO but I am unable to find such an exapmle. Can anybody please help ...
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### Divergent sequence with decreasing function

Let $f: \mathbb{R} \to (0,\infty)$ be a decreasing function. Define a sequence $(a_n)$ by $a_1=1$ and $a_{n+1}=a_n+f(a_n)$ for every $n\ge 1$. Prove that $(a_n) \to \infty$. I have tried by ...
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### Sum of the distance between consecutive terms implies the sequence is Cauchy.

If a sequence $(x_n)_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum_{n\geq 1} ||x_n-x_{n+1}||<\infty$, show that it is Cauchy. This isn't a complete answer, but here's my train of thought. ...
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 ...
Is it sufficient to show that for any $\epsilon > 0$ that there exists $N$ such that if $n$ is greater than or equal to $N$, then $d(s_n, s_{n+1})$ is less than $\epsilon$ to prove that a sequence $... 2answers 114 views ### How to show that$x_n=(1+\frac{1}{n})^n$is a cauchy sequence in$\mathbb Q$. [duplicate] How to show that$x_n=(1+\frac{1}{n})^n$is a cauchy sequence in$\mathbb Q$. I am trying to use binomial theorem to expand$(1+\frac{1}{n})^n$but it is not coming. 4answers 39 views ### Convergence of Cauchy's sequence I understood that every convergent sequence is a Cauchy sequence. It seems that the converse is not necessarily true. An example given is the sequence$\{x_n\}$, where$x_n = (0.1)^n$is a Cauchy ... 3answers 76 views ### How to prove that$\left\{\frac{1}{n^{2}}\right\}$is Cauchy sequence How can I prove that$\left\{\frac{1}{n^{2}}\right\}$is a Cauchy sequence? A sequence of real numbers$\left\{x_{n}\right\}$is said to be Cauchy, if for every$\varepsilon>0$, there exists a ... 2answers 90 views ### How to show that$\sin(n)$does not converge ONLY by using Cauchy's criterion? I know this question has been asked before... I went through all of the questions of this sort and none of them had an answer using Cauchy's criterion. I know that$\sin(n)$does not converge and I ... 1answer 159 views ### Question about Cauchy Sequence proof for$\sum_{i=1}^\infty \frac{1}{i^2}\$
I was looking through a proof that the sum of $$\sum_{i=1}^\infty \frac{1}{i^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}...$$ is convergent. I know there are probably other ways to prove this, ...