For questions relating to the properties of Cauchy sequences.

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2
votes
4answers
219 views

Why are all convergent sequences necessarily Cauchy?

I can understand the proof, which I could do myself: $|s_n - s_m| = |s_n - s + s - s_m|$ $\Rightarrow |s_n - s_m| \leq |s_n - s| + |s_m - s| $ For some $\epsilon > 0, \exists\ \ N(\epsilon) ...
2
votes
1answer
29 views

Cauchy sequences are bounded in every metric space

A few days laid out an example, and asked for help, and @ shadow10 replied, his answer the question of can I find the Every Cauchy sequence is bounded but please someone help me in relation to ...
0
votes
1answer
25 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit ...
0
votes
1answer
49 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
1
vote
1answer
28 views

How to prove the 'uniform summability' of a Cauchy sequence?

I have an exercise given by the teacher and I'm pretty sure that this proof is not hard, but I don't have idea how to approach it. I have to prove the 'uniform summability' (this name was used by ...
0
votes
1answer
24 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
0
votes
1answer
79 views

Every Cauchy sequence is bounded

Please help me to understand step-by-step how this example is proven. The statement is follows: Every Cauchy sequence is bounded. since I do not understand how verified, please help me, thank ...
1
vote
1answer
23 views

Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
1
vote
5answers
99 views

Show that $\langle f_n \rangle$, where $f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$ is a Cauchy sequence.

Show that $\langle f_n \rangle$, where $$f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence. My attempt: Consider $$|f_{2m}-f_m| = \left| ...
1
vote
2answers
35 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
votes
4answers
339 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
votes
2answers
70 views

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

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
votes
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
votes
1answer
29 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}}, ...
1
vote
3answers
76 views

prove ${a_n}$ is a Cauchy sequence, provided $a_{n+2} = \frac{a_n + a_{n+1}}{2}$ [closed]

Suppose there is a sequence with the property $$a_{n+2} = \frac{a_n + a_{n+1}}{2}$$ for all $n \in \mathbb{N}_{+}$ Prove that ${a_n}$ is a Cauchy sequence I've self-taught ...
0
votes
1answer
75 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
votes
2answers
33 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
votes
1answer
67 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
54 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 ...
1
vote
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
votes
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
52 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
votes
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
vote
1answer
15 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
votes
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
votes
0answers
29 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}, \ ...
1
vote
1answer
45 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
votes
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 ...
5
votes
2answers
145 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 ...
-1
votes
2answers
34 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
votes
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
votes
3answers
71 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
99 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 $\int_{-\infty}^{+\infty}f$ ...
1
vote
1answer
58 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
votes
1answer
176 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?
1
vote
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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votes
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
vote
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
votes
2answers
57 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
vote
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
vote
2answers
56 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 $|x−s_N| < \epsilon$
1
vote
1answer
34 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
votes
1answer
32 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
vote
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
votes
1answer
44 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 $|a_{m}-a_{n}| < \epsilon$ ...
0
votes
0answers
29 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 ...