# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by

Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by $\beta _n = (\int_E |f_n|^pd \mu)^{\frac{1}{p}}$ and suppose that $f_n$ is a ...
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### If $\sum x_n$ converges, is the even partial sums of the sequence squared Cauchy?

If $\sum x_n$ converges, is the sequence $a_k =\sum_{n=1}^k x_{2n}^2$ Cauchy? I suspect that it isn't but a bit stuck on finding a counter example. For a valid counterexample, I believe I need a ...
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### Understanding a Cauchy Sequence Proof

I've got some general confusion over Cauchy sequences I'm trying to clear up. Intuitively, I know that the Cauchy theorem lets us prove convergence of a sequence without knowing about the real number ...
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### Show that a recursive sequence $(x_n)$ is Cauchy

The given sequence is defined as $x_1 = 0$, $x_2 = 1$ and $x_{n+2} = \frac{1}{3}x_{n+1} + \frac{2}{3}x_n$ for $n \geq 1$. I seek to show that it is Cauchy. So how I planned on showing this was to ...
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### Determine if subset $S$ of $\mathbb{R}$ is complete

Given $S = \{x \in \mathbb{R} \setminus\mathbb{Q}\mid x^2 \leq \frac{1}{4} \} \subset \mathbb{R}$, determine if it is complete. Given that a subset $S$ of $\mathbb{R}$ is $\textit{complete}$ if ...
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### Prove the sequence is Cauchy

Prove with definition that it is Cauchy $$a_n = \frac{n+3}{2n+1},$$ wheree $n$ is a natural number I have seen other examples such as $\frac{1}{n}$ and such that show how to prove they are Cauchy ...
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### For a sequence, how should we check that the distance between two consecutive terms $a_{n+2}$ and $a_{n+1}$ is less than $a_{n+1}$ and $a_n$?

Background It can be proved that for $0 < \alpha <1$, if $\{a_n\}$ is a sequence which satisfies $$|a_{n+2} - a_{n+1}| \leq \alpha |a_{n+1} - a_n|$$ Then $a_n$ is a Cauchy ...
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### What sequences are Cauchy in all metrics for a given topology?

Different metrics for the same topology can have different sets of Cauchy sequences. But I'm interested in what sequences are Cauchy in every metric for a given topology. For a completely metrizable ...
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### Why can't completeness be defined on topological spaces without using metrics?

I have heard it said that completeness is a not a property of topological spaces, only a property of metric spaces (or topological groups), because Cauchy sequences require a metric to define them, ...
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### Let $s_{0}$ and $s_{1}$ be arbitrary and $s_{n+1}$=($s_{n}$+ $s_{n-1}$ ) /2 n>=1 show Cauchy [duplicate]

Let $s_{0}$ and $s_{1}$ be arbitrary and $s_{n+1}$=($s_{n}$+ $s_{n-1}$ ) /2 n>=1 show Cauchy show {$s_{n}$} is a Cauchy sequence. Conclude that {$s_{n}$} is convergent.
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### Cauchy sequences - Infinite sums

Let $\{x_n\}$ be a sequence of real numbers. Suppose that for each $\epsilon > 0$ there is an $N \in \mathbb{N}$ such that $m \geq n \geq N$ implies $|\sum_{k=n} ^m x_k | < \epsilon$. Prove ...
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### If for all $n \in N$, $|a_n| < 2$ and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ .

Suppose that ${a_{n}}$ is a sequence such that, for all $n \in N$, $|a_n| < 2$, and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ . Prove that ${a_n}$ is a Cauchy sequence. My ...
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### Is the sequence $a_{n} = \frac {1}{2^2} + \frac{2}{3^2} + … +\frac{n}{(n+1)^2}$ Cauchy?

First: I tried substituting natural numbers for $n$ to calculate the consecutive terms of the sequence and then see the difference between their values and I found that the difference is decreasing ...
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### Counterexample: Cauchy condition without absolute value for convergence of a sequence

Let $(a_n)$ be a sequence of non-negative real numbers, not necessarily increasing. The Cauchy criterion states that $(a_n)$ converges if and only if $\forall \epsilon >0$, there exists an $N$ ...
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### Cauchy sequence where the set of outputs is finite

If $\{a_n\}$ is a Cauchy sequence, and $S = \{a_n |n\in\mathbb{N}\}$ is finite, then $\{a_n\}$ is constant from some point on. The statement makes sense, but I'm not quite sure how to start. I feel ...
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### Cauchy sequence exercise question (Exercise 2.6.4 Abbott analysis)

I have a problem solving the question below. I'm stuck at the stage (ii) of my solution below. Problem: Let $(a_n)$ be a Cauchy sequence. Decide whether the following sequence is a Cauchy sequence, ...
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### There are no nontrivial metric spaces such that every sequence is a Cauchy sequence

Is my intuition correct here? If our metric space consists of a single point $x$, any sequence in it would just be $x, x, x, \dots$ which is obviously Cauchy. If we have any metric space $(M, d)$ ...
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### Proving that a sequence is not Cauchy

I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy ...
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### Proving this sequence is Cauchy

How can I prove that the sequence $\{x_k\}$ where $x_1 = 4, x_k = x_{k-1} + 4(\frac{1}{4k-3} - \frac{1}{4k-5})$ is Cauchy? I understand the definition of Cauchy and I can see that the sequence is ...
### Let $0 < a \leq 1$ and $s_1 = a/2$ , $2s_{n+1} = s_n^2 + a$ , Then how to show that the sequence is convergent.
Let $0 < a \leq 1$ and $s_1 = a/2$ , $2s_{n+1} = s_n^2 + a$ , Then how to show that the sequence is convergent. My Try : I have tried to find out $s_{n+1} - s_n$ and try to understand the ...