# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Cauchy sequence such that don't have limes in C[0,1]

Give an example of series $f_n \in C[0,1]$ such that $f_n$ is Cauchy sequence in norm $$\|(a_n)\|_p = \left( \sum_{n=1}^{\infty} |a_n|^p \right)^{1/p}$$ and $$\lim_{n \to \infty} f_n(x)$$ don't ...
3answers
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### Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t.$ Is this space a Hilbert ...
3answers
478 views

### Is $x_n=\frac{1}{n}$ Cauchy sequence with the metric $d(x,y)=|\arctan x−\arctan y|$? [closed]

Let $(\Bbb{R},d)$ be a metric space where $d(x,y)=\left\vert\arctan x−\arctan y\right\vert$. Is the sequence $x_n=\frac{1}{n}$ a Cauchy sequence with this metric? The definition of Cauchy sequence ...
2answers
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### Examples of Sequences

Can someone give examples for these types of sequences? A sequence that is monotone but not convergent A sequence that is not bounded but is convergent A sequence that is monotone but not Cauchy A ...
2answers
91 views

### Prove $\sum_{n = 1}^{\infty} 2^{-n} x^n$ does not converge uniformly on $(-2, 2)$

How can one go about proving this? (I understand that the said series does converge uniformly on all $[-a, a]$ where $0 \leq a < 2$.) I am especially interested in knowing if there is a way to ...
3answers
146 views

### Show something is a Cauchy sequence

If I want to show that something is Cauchy, should I show it converges and then show it is Cauchy or should I go at it straight from the definition. I am just trying to figure out generally what to ...
3answers
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2answers
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### What's wrong with the classical Cauchy construction of the reals?

I am reading Bishop's "Constructive Analysis" and he says that defining a real number to just be an equivalence class of Cauchy sequences of rationals would be wrong. Why is that?
2answers
116 views

### What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean. What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a ...
1answer
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1answer
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### bounded sequence problem

$(a_n)$ is a positive sequence such that: $$\forall \epsilon > 0 ,\quad \exists N\in \mathbb{N}, \quad \forall n,m >N ,\quad |\frac{a_{n}}{a_{m}} - 1|< \epsilon$$ how to prove this ...
1answer
69 views

### Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
2answers
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1answer
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### Proof that this sequence is Cauchy

How to prove that if ${a_n}$ is a Cauchy sequence, if $x_n$ is a sequence and there is a constant $C$ such that, $$|x_n-x_m| \le C|a_n-a_m|$$ Then $x_n$ is also a Cauchy sequence. Any hints or ...
3answers
613 views

### Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.

Let $(x_n)_{n=1}^{\infty}$ be a sequence such that $|x_{n+1} - x_n| < r^n$ for all $n \geq 1$, for some $0 < r < 1$. Prove that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence. I understand that ...
3answers
1k views

### Find a limit of the recursive sequence

The task is to prove sequence convergence and find a limit. $x_0=0$ $x_1=1$ $x_{n+1}=\frac {x_n + n \cdot x_{n-1}} {n+1}$ I have computed some values of a sequence to build up some idea of the data: ...
1answer
277 views

### Proof That a Sequence is Cauchy [duplicate]

$\{a_n\}$ is a sequence such that $|a_{n+1} - a_n| < 5^{-n}$ for all $n>0$. How to prove that this sequence is a Cauchy sequence?
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
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1answer
372 views