For questions relating to the properties of Cauchy sequences.

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2answers
31 views

Does a bounded continuous function map Cauchy sequences to Cauchy sequences?

I only ever see the example of $f:(0,1]\rightarrow \mathbb{R}$ where $f(x)=\frac{1}{x}$as that of a continuous function that does not map Cauchy sequences to Cauchy sequences. Are there examples of ...
7
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5answers
331 views

Cauchy sequences - can we control the rate at which elements “get closer”?

In Simon & Reed's book Methods of Modern Mathematical Physics, it is proven in chapter 1 (Theorem 1.12) that $L^1$ is complete (Riesz-Fisher theorem). The proof starts off as follows: Let ...
-1
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2answers
66 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [on hold]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
0
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0answers
37 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ ...
0
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1answer
27 views

Proof sequence is cauchy

Prove that a sequence satisfying: $|x_k-x_{k+1}|< \frac{1}{a^k}$ for some $a>1$, for all $k$ is cauchy. It follows from the criterion that: $|a_n - a_m|<\sum_{i=n}^{m}{\frac{1}{a^i}}, ...
0
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1answer
71 views

Prove $a_{n+1} = \frac{4+3 a_n}{3+2 a_n}$ is a Cauchy sequence [closed]

How to prove that a sequence $a_n$ as defined $a_{n+1} = \frac{4+3 a_n}{3+2 a_n}$ is a Cauchy sequence?
2
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2answers
28 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
3
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1answer
62 views

Limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ - help finishing proof…

EDIT: Fixed geometric proof off-by-one error. I am looking for the limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ with $x_2 > x_1$ arbitrary. I can show $\forall n: |x_{n}-x_{n+1}| = \frac{c}{2^{n-1}}$ ...
3
votes
1answer
51 views

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 under ...
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0answers
30 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
0
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1answer
41 views

Determine whether $f_n(x) = \frac {nx}{1+(nx)^2}$ is cauchy in $[ C^0([−1, 1], \mathbb {R} ), d_\infty]$

I have a homework question: Is the sequence $$ f_n(x) = \frac {nx}{1+(nx)^2} $$ Cauchy in the space $ C^0([−1, 1], \mathbb {R} ) $ with the metric induced from the sup norm? Could you please write ...
3
votes
1answer
50 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
0
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3answers
77 views

Let $a_n=\frac{a_{n-1}+a_{n-2}}{2}$ for each positive integer $n\geq 2$. Show that $\{a_n\}_{n=1}^{\infty}$ is Cauchy

Let $a_0$ and $a_1$ be distinct real numbers. Define $a_n=\frac{a_{n-1}+a_{n-2}}{2}$ for each positive integer $n\geq 2$. Show that $\{a_n\}_{n=1}^{\infty}$ is a Cauchy sequence. Hint: You may want ...
1
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1answer
14 views

Net-Complete $\iff$ Sequence-Complete

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
2
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2answers
42 views

Convergence of series $\sum a_rb_r$

Assuming $\sum a^2_r$ and $\sum b^2_r$ converge, can we deduce that $\sum a_rb_r$ converges? It feels like we can, but how? Using Cauchy Criterion for convergence maybe? Can you hint me? Thanks a ...
0
votes
0answers
28 views

Proving a Cauchy Sequence without an explicit formula for $(a_n)$ [duplicate]

How would I prove that for all $n$ in the natural numbers, $|a_{n+1} - a_n| \le 1/2^n$ $(a_n)$ is a Cauchy Sequence? I have tried to use the standard definition and the triangle inequality, but ...
0
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0answers
28 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ ...
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1answer
44 views

Imaginary unit $i$ is not a limit of a real Cauchy sequence

I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued ...
0
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1answer
27 views

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 ...
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2answers
140 views

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 ...
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2answers
33 views

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 ...
0
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2answers
60 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 ...
0
votes
3answers
32 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 ...
-1
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3answers
70 views

How many real valued Cauchy sequences are there? [closed]

Is the set of all Cauchy sequences of real numbers countable or uncountable? In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert ...
2
votes
4answers
97 views

Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.

Consider the distance function given by $d:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\;d(x,y)=|\int_x^yf(t)dt|$ where $f$ is a continuous and positive function such that $\int_{-\infty}^{+\infty}f$ ...
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1answer
53 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
6
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1answer
172 views

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?
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2answers
77 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 ...
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1answer
39 views

prove that $x_n=\frac{1}{n}$ is cauchy sequence [closed]

Could you please prove that $x_n=\frac{1}{n}$ is a cauchy sequence? Thanks a lot.
1
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1answer
26 views

On the cauchy sequence $x^n$ in $(C[0,1],\|\cdot\|_\infty)$

I know how to prove that $x^n$ is not cauchy in $(C[0,1],\|\cdot\|_\infty)$, but my question is that since $(C[0,1],\|\cdot\|_\infty)$ is complete, and the pointwise limit on $x^n$ is: $x^n\to f(x) = ...
2
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2answers
51 views

Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
1
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1answer
16 views

negative cauchy function

attempt: from defn of uniformally continuous $\forall \epsilon > 0 $ $\exists \delta > 0 $ s.t. $|x-y| < \delta $ with $x,y \in (-\infty,0)$ $\implies |f(x) - f(y)| < \epsilon$. My idea ...
1
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2answers
54 views

Cauchy sequence and uniform continuity

I read somewhere that because uniform continuous function maps Cauchy sequence to Cauchy sequence and Cauchy sequence is bounded, so the function must be bounded. I am not sure if it is correct. My ...
0
votes
2answers
26 views

Construct a sequence with certain property

Construct a sequence $(s_n)$ which satisfies the following property: $\forall x \in \mathbb{R}$ and $\epsilon > 0$, there exists some $N$ such that $|x−s_N| < \epsilon$
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1answer
33 views

Is the set of all integers with metric $d(m,n)=|m-n|$ a complete space?

Consider the set of integers with a metric defined by $d(m,n)=|m-n|$.Is this set complete with respect to this metric? If it is a metric, then I am stuck here. How can a Cauchy sequence have a limit ...
0
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1answer
29 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
1
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0answers
110 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, ...
0
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1answer
43 views

If a sequence converges, then it is Cauchy?

Considering the following proof and its converse: If a sequence converges, then it is Cauchy. That is, if $\lim_{n\to \infty}a_{n} = L$, then given $m>N$, we have that $|a_{m}-a_{n}| < \epsilon$ ...
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0answers
28 views

Is this a valid proof of Mertens summation theorem?

Consider $\sum a_n$ absolutely convergent and $\sum b_n$ convergent. Then $\sum c_n$ with $c_n = \sum_{k=0}^{\infty} a_n b_{n-k}$ is convergent, too. So my proof attempt is a little short and I am ...
0
votes
2answers
33 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
1
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2answers
162 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
0
votes
1answer
23 views

Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
2
votes
2answers
88 views

Showing a sequence is Cauchy

Let $f(x)$ be a function which is differentiable on $\mathbb{R}$ and for which $a = \sup\{|f ′(x)| : x ∈ \mathbb{R}\}$ is less than 1. Let $x_{0}$ be any fixed real number. (a) Show that the sequence ...
1
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1answer
42 views

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 ...
0
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0answers
34 views

metric makes $\mathcal{C}(\Omega) $ to a complete metric space

I have some problem with the second part Show that $$\rho(f,g):= \sum_{m=1}^\infty \frac{1}{2^m} \frac{\max_{x \in \bar{\Omega}_m}|f(x)-g(x)|}{1+\max_{x \in \bar{\Omega}_m}|f(x)-g(x)|}$$ is a metric ...
0
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1answer
41 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 ...
2
votes
2answers
79 views

Prove ${a_n} = \lim_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria

Prove ${a_n} = \lim\limits_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria. What I did: Let $n, m=n+k \in \mathbb{N}$. $$\left| {{a_m} - ...
3
votes
5answers
483 views

What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
0
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1answer
27 views

Completeness of special set

Let C denote continuous bounded functions from R -> R which are identically zero outside some closed bounded interval. Is C complete in the sup norm? I think it is not because there will be some ...
1
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1answer
64 views

Is the space of continuous functions a Cauchy complete?

I am so new to functional analysis so I am looking for an answer of a confusion I am having right now in my mind because I have seen many different answers for the question I am gonna ask below. I ...