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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

A sequence $\{x_n\}$ in an arbitrary metric space, and in particular the space $\Bbb{R}$, is called Cauchy if the terms of the sequence become arbitrarily close together; that is, for every $\epsilon > 0$, there exists an $N$ such that

$$n, m \ge N \implies d(x_n, x_m) < \epsilon$$

where $d$ is the distance function for the metric space. In the particular case of the real numbers, this condition becomes

$$n, m \ge N \implies |x_n - x_m| < \epsilon$$

A complete metric space is a metric space in which every Cauchy sequence is convergent; this gives an alternate definition of convergence of a sequence that does not rely on the limiting value.

Source: the Cauchy sequence article on Wikipedia.

2431 questions
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Why is the sequence $x(n) = \log n$ **not** Cauchy?

I read in the book Applied Analysis by Hunter and Nachtergale that the sequence $x(n)=\log(n)$ is not Cauchy since $\log(n)\to\infty$ But that seems to be irrelevant to the definition of a Cauchy sequence which I understand is as follows: A…
Alhpa Delta
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A sequence whose sum is infinite but whose sum of squares is not?

I am thinking of positive sequences whose sum is infinite but whose sum of squares is not? One representative sequence is $$x[n] = \frac{a}{n+b},$$ where $a$ and $b$ are given real numbers such that $a>0$ and $b\ge0$. I know that there will be…
Danny_Kim
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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 for all $n>N$, $|a_{2n+1}|, |a_{2n}| < \epsilon$. As…
user84899
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Cauchy sequence $x_n=\sqrt{a+x_{n-1}}$

I have to show that this sequence $$ x_n=\sqrt{a+x_{n-1}} \hbox{ with } x_1=\sqrt{a} $$ is a Cauchy sequence for every $a>0$. I have done the following calculations: $$ \left| x_{n+2}-x_{n+1} \right|=\left|…
Claudio
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How can one know every Cauchy sequence in a complete metric space converges?

I am new to Cauchy sequences. I stumbled onto them in the process of learning what a Hilbert space is. As I understand it, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. But…
Stan Shunpike
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Show that $(0,1)$ is completely metrizable

Here, under the Section 'Examples ', $(0,1)$ is not complete with its usual metric inherited from $\mathbb{R}$, but it is completely metrizable since it is homeomorphic to $\mathbb{R}$. To show $(0,1)$ is not complete with its usual metric, define a…
Idonknow
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Is this proof about Cauchy sequence correct?

Let $(x_n)$ be a Cauchy sequence in R and $(y_n)$ is a sequence in R such that $|x_n-y_n|<\frac{1}{n}$ for all $n\geq 1$. Prove that $y_n$ is a Cauchy sequence and $lim (x_n)=lim (y_n)$ My attempt (rough sketch): Assume $y_n$ is a sequence in R. If…
Natasha J
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How to prove that the following is Cauchy?

So I have to prove that the following is a Cauchy sequence by definition: $|u_{n+2}-u_{n+1}|\leq \frac{1}{2}|u_{n+1}-u_n|$. I know that, $$|u_{n+2}-u_{n+1}|\leq \frac{1}{2^n} |u_2-u_1|$$ How do I proceed further?
Natasha J
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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 for large values of $n$ (not very large because I…
Intuition
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Prove $a_n$ is a Cauchy-sequence, with $a_0 \in \Bbb R$ and $a_{n+1}=f(a_n)$

Let $f: \Bbb R \to \Bbb R$ be a differentiable function with $m=sup${$|f'(x)||x \in \Bbb R$} $<1$. Let $a_0 \in \Bbb R$ and define $a_{n+1} =f(a_n)$ for $n=0,1,2...$. Prove the sequence $(a_n)_{n \geq 0}$ is a Cauchy sequence. So we have to prove…
Fruitdruif
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(verification) (M,d) be a metric space with every bounded sequence has a convergent subsequence $\Rightarrow$ M is complete

Claim (M,d) be a metric space with a property that every bounded sequence has a convergent subsequence $\Rightarrow$ M is complete Proof To be complete we need to verify $\forall$ cauchysequence in $M$ converges in $M$. first what we know is that…
Daschin
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$\left|x_{n+1} - x_n\right| < C\left|x_n - x_{n-1}\right|$ prove this is Cauchy

Suppose that for the sequence $\{x_n\}$ there exists $0 < C < 1$ such that $$\left|x_{n+1} - x_n\right| < C\cdot \left|x_n - x_{n-1}\right|.$$ Prove that ${x_n}$ is Cauchy.
TAPLON
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Is $ \frac{1}{2\sin(n)+0.5\sqrt{n}+1.1}+n$ a Cauchy Sequence?

Is $a_n= \dfrac{1}{2\sin(n)+0.5\sqrt{n}+1.1}+n$ a Cauchy sequence? If I plot the difference between $a_n$ and $a_{n+1}$ it seems to be shrinking as $n$ goes to infinity, with the difference reaching a limit of 1.
Gustavo
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Metric space -completeness

Show that $(C[0,1],\|\cdot\|_2)$ is not complete. For that consider the sequence $(g_n)_{n\in \mathbb{N}}$ defined below and show that this is not a convergent Cauchy-sequence. $g_n:[0,1]\to \mathbb{R},~g_n(t):=\begin{cases}f_n(t)~~~~~~~ t\in…
Nicole Parson
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Is this enough for a epsilon N proof? Cauchy sequence

Assume that $ C \in \mathbb{R}, q < 1$ and $|x_{n+1} - x_n | < Cq^n$ for all n. Prove that this is a Cauchy-sequence... What I do is that I assume without loss of generality that $m < n$ so $n = m + k$. Then by the triangle inequality, $$|x_{m+k} -…
Marcus Ma
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