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 ...
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 $\int_{-\infty}^{+\infty}f$ ...
1
vote
1answer
16 views

Help undestanding compactness with convergent subsequences

One way to define compactness in metric spaces is to note that in compact metric space each sequence has a convergent subsequence. Understanding compactness is difficult for me from this ...
1
vote
1answer
53 views

Proving that a set of metric space is dense in $A$ iff there exists a sequence converging to $x\in A$

I'm using the following definition: A set $M$ of a metric space $(\frak M,\rho)$ is called dense in a set $A\subset\frak M$ if $$\forall \varepsilon>0,x\in A\exists y\in ...
0
votes
1answer
46 views

Show these sequences converge and determine the limit of each.

Using the definition of convergence in metric spaces, show that the following sequences converge and find its limit. 1.) $a_n(x)=\frac{n}{n+1}x^2+\frac{2}{n}x+3$ in $(C[0,1],||.||_1)$ To begin we ...
0
votes
1answer
33 views

Show that $d_g$ is a metric on $l^1$.

On the space $l^1$ of complex valued sequences $(x_n)$ such that $\sum|x_n|<\infty$, define for $x=(x_1,x_2,\cdots)$, $y=(y_1,y_2,\cdots)$ the metric $d_f$ by ...
0
votes
0answers
53 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
0
votes
3answers
45 views

Is the convergence of a sequence independent of the chosen metric?

Given a metric $\rho$ on $X$ and a sequence $x_n$ in $X$. Does the convergence of $x_n\to x$ under $\rho$ also imply the convergence to the same limit under any other metric $\sigma$? I don't know th ...
0
votes
2answers
60 views

Uniform convergence and pointwise convergence

Having the standard definitions for pointwise resp. uniform convergence of sequences of functions in a general metric space ($X,d$). What special conditions should X fulfill such that pointwise ...
4
votes
1answer
45 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
2
votes
1answer
65 views

Cauchyness vs. Double Limits

Maybe there are some textbooks which might treat cauchyness by taking double limits... My question: Is it sufficient and necessary to consider the double limit: $$x_n\quad \text{cauchy}\quad ...
1
vote
1answer
24 views

Convergence of the limits of a double sequence in one variable as a sequence of the other variable

If $a_{n,m}$ is a double sequence in a metric space such that $a_{n,m} \rightarrow_m a_n$ uniformly on $n$ and $a_{n,m} \rightarrow_n a$ for all $m$, then $$a_{n} \rightarrow a.$$ Indeed for any ...
0
votes
1answer
63 views

Prove that the space of divergent sequences in $(l_{\infty},d_{\infty})$ is open and dense. Is it separable?

The problem statements are: Consider the space $A=\{ \{a_n\}_{n \in \mathbb N} \in l_{\infty} : \{a_n\}_{n \in \mathbb N} \text { is not convergent }\}$ $a)$ Prove that $A$ is open and dense in ...
1
vote
0answers
42 views

Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
2
votes
1answer
85 views

uniformly convergent sequence of functions on a compact space

There's an exercise on Kaplansky's textbook that says: Let $\{f_i\}$ and $f$ be continuous real functions on a compact metric space $M$. Prove that $f_i$ converges uniformly to $f$ if and only if the ...
0
votes
0answers
160 views

Proving existence of uniformly convergent subsequence of a sequence of functions

Let $\{f_n\}_{n \in \mathbb N}$ a sequence of integrable and uniformly bounded functions $f_n:[a,b] \to \mathbb R$ and for each $n$ let $F_n:[a,b] \to \mathbb R$ such that $F_n(x)=\int_a^x f_n(t)dt$ ...
5
votes
3answers
159 views

Does $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ imply anything?

Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not ...
1
vote
2answers
339 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
4
votes
1answer
290 views

Existence of limit for convergence by measure for Cauchy-in-measure sequence+completeness of metric space?

Sorry if this is the wrong place to put it. But this question come from a graduate level textbook and seems pretty hard to me, so I hope this is a good place. Anyway, this come from the book Real ...
0
votes
1answer
52 views

prove that metric and series

Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| \leq 1$ for every positive integers $n$.Let $\{a_n\},\{b_n\} \in E$ Prove that ...
4
votes
1answer
115 views

topology homework

I'm new to topology, and therefore not very good at it yet. I have following questions, that I have ansewer, please help me verify what is not correct and what is missing in my answers. Let $X$ be ...
4
votes
2answers
233 views

Bounded derivative implies bounded function?

By the following theorem, it suffices to show that $\{F_n: n\in\mathbb N\}$ is equicontinuous and bounded: If $f_k$ is a sequence in an equicontinuous and pointwise bounded set of maps from a ...
1
vote
1answer
84 views

Property of a metric in the space of all the sequences of real numbers

A few weeks ago I had this problem, the adjoint-teacher solved it on class, and I thought I understood, but now I'm rechecking and there are a few things that aren't clear for me. So we defined this ...
1
vote
1answer
77 views

A question on Cauchy sub-sequences in a metric space $(X,d)$

Let $(X,d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of positive $\epsilon_k \rightarrow 0$, there is a ...
2
votes
3answers
92 views

$x_1=0,\,x_{2n}=\frac{x_{2n-1}}{2},\,x_{2n+1}=x_{2n}+\frac{1}{2}$ Find $\lim \sup {x_n}$ and $\lim \inf {x_n}$

Define a real sequence recursively by the following equations: $$x_1=0$$ $$x_{2n}=\frac{x_{2n-1}}{2}$$ $$x_{2n+1}=x_{2n}+\frac{1}{2}$$ for each $n \in \mathbb{N}$. Find $\lim \sup {x_n}$ and $\lim ...
3
votes
1answer
126 views

How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$

I have to prove that this is a metric: $$d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$$ The only thing that I can't prove is the triangle inequality, it's really hard. I know that ...
2
votes
0answers
291 views

Is this proof that every convergent sequence is bounded correct?

I've tried the following proof that a convergent sequence is bounded but I'm not sure if it is correct or not. Let $(M,d)$ be a metric space and suppose $(x_k)$ is a sequence of points of $M$ that ...
1
vote
1answer
727 views

If every convergent subsequece converges to the same limit then the sequence converges

I came across this question: In a compact metric space $(X,d)$ if every convergent subsequence of a sequence converges to the same limit, say $l$, then the original sequence also converges to $l$.
2
votes
0answers
71 views

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene $x\sigma=(x_{\sigma_n})$ where $\sigma:\mathbb N\to\mathbb N$ is strictly increasing. Then $(x_n)$ is ...
2
votes
1answer
355 views

Justify: In a metric space every bounded sequence has a convergent subsequence.

Justify: In a metric space every bounded sequence has a convergent subsequence. My Attempt: False: Consider the metric space $(X,d)$ where $X=\mathbb R$ and $d$ is the discrete metric on $X.$ ...
0
votes
1answer
83 views

Extension of Cauchy sequentially regular function

To prove: If $A$ is a subset of a metric space $(X,d)$ and there is a function $f$ from $A$ to a complete metric space $(Y,e)$ which maps Cauchy sequences to Cauchy. Then there exists a unique ...
0
votes
3answers
116 views

Function to uniquely map a set of rectangles in space to a number?

I am trying to build a new way of indexing spatial data. Is there a function that takes as parameter a number of rectangles in euclidean space, and outputs an unique number?Can such a function be ...
2
votes
1answer
62 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
1
vote
1answer
85 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
1
vote
2answers
83 views

If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.

A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$ Let $(X,d)$ be a metric space and let ...
1
vote
1answer
33 views

Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?

For example, suppose $$ x_n = \frac 1n \\ X = (0, 1)$$ Is $x_n$ convergent in $X$? My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
2
votes
4answers
518 views

Understanding Complete Metric Spaces and Cauchy Sequences

From my own definition, I have concluded that a complete metric space is a set and a metric where the set consists of no holes in it. Book definitions describe that "A complete metric space is a ...
1
vote
1answer
93 views

About Convergence of the Image of a Convergent Sequence Under a Uniformaly Convergent Sequence of Functions

Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose ...
1
vote
1answer
63 views

Sequences of functions

Suppose that $X$ is a compact metric space. Let: (a) $(f_n)$ be a sequence of real-valued continuous functions on $X$ (b) $(f_n)$ converges pointwise to a continuous function $f$ on $X$ (c) $f_n(x) ...
7
votes
2answers
334 views

“The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$” and its compactness

[I really want to apologize if this problem looks a little too long.] The problem : This is taken from here [Question. 3.7 (c)] and it says... Prove or disprove the comapctness of the closure of ...
1
vote
1answer
355 views

Proof about diameter of a set

I could not prove the following question could you please help me? Best Regards Let $X, d(x, y)$ be a metric space. By definition, diameter of a bounded set $A ⊂ X$ is the number $diam(A)$ = ...
2
votes
3answers
129 views

Every sequence is composed of isolated points?

Let $(M,d)$ be a metric space and $\{x_n\}_{n=1}^\infty\subset M$ be a sequence. Prove that $$\forall n\in\mathbb N,\quad\exists \varepsilon> 0 \;B(x_n,\varepsilon)\cap \{x_n\}_{n=1}^\infty = ...
2
votes
1answer
55 views

Metric space and sequences

Consider the metric space $\langle \mathbb I,d\rangle$ where $\mathbb I$ is the set of all irrational numbers, and $d$ is the usual distance metric. For each $n\in\mathbb Z^+$, let $x_n =\frac{n + ...
0
votes
2answers
97 views

Two proofs about metric spaces and one about series

In a few hours I will have a quiz and while studying I had some questions. Could you please help me? Thanks in advance Question 1: Let $(X, d)$ be a metric space and $(x_n)$ be a sequence in ...
0
votes
2answers
173 views

Coinciding open sets

I'm given two distances that are defined on some metric space and I need to show that open sets and Cauchy sequences coincide for the two distances. What does this mean? I'm avoiding giving details on ...
2
votes
4answers
506 views

Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$? The pointwise limit is not ...
3
votes
1answer
195 views

Compact subsets in $l_\infty$ (converse of my last question)

(Converse of my last question) If $A \subseteq \ell_\infty$, and $A=\{l\in \ell_\infty: |l_n| \le b_n \}$, where $b_n$ is a sequence of real, non-negative numbers, then if $\lim (b_n) = 0$ it must ...
2
votes
0answers
114 views

Prove metric space…

Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by $$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$ Prove that $d$ is ...
3
votes
1answer
95 views

Compact subsets in $l_\infty$

If $A \subseteq l_\infty$, and $A=\{l\in l_\infty: |l_n| \le b_n \}$, where $b_n$ is a sequence of real, non-negative numbers, then if $A$ is compact subset of $X$ it must mean that $\lim (b_n) = 0$. ...
3
votes
1answer
133 views

Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...