# Tagged Questions

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### What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if$x <1/n$}\\ 1 \ \mbox{ if$x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if$x ...
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### Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
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### Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
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### semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
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### 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!)
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### How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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### Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $A$ of a metric space and its limit point $a$, we know that for every $\epsilon > 0$ there is another point $x$ different from $a$ ...
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### Completeness proof?

First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is ...
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### 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 ...
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### Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
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### 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 ...
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### Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
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### Convergent sequence in product space on $\mathbb{R}^{\omega}$

I am confused about the concept of convergent sequence in product space when learning Munkres's Topology, especially when I am comparing two related exercises of it. The exercise 6 of section 19 ...
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### Describe all the convergent and Cauchy sequences in this metric space

Consider the set of natural numbers $\mathbb N$ with the metric $$d(m,n)=\frac{\left|m-n\right|}{1+\left|m-n\right|}$$ Describe all convergent sequences and all Cauchy sequences in this metric ...
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### Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...
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### Countable sum of closed boundary sets

I have to prove that for complete metric space and $f_n$ converge pointwisely to $f$ $f^{-1}(a,b)\setminus Int(f^{-1} (a,b))$ is countable sum of closed, boundary sets. Here is my solution: ...
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### On eventually constant sequences

It is of course true that in a discrete space a sequence converges iff it's eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the ...
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### the power series converges in compact convergence topology

Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$ a) Prove that $(f_{n})$ converges in the topology of compact convergence, ...
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### 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 ...
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### Check the convergence of a sequence

Let $X$ =$[0,1]$ and $d(x,y)=|x-y|/(1+|x-y|)$ be the metric defined on $X$. Then check whether the sequence ${x_n = 1/n^2}$ A) Converges in $(X,d)$ B)Does not converge in $(X,d)$ My attempt : I ...
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### Closure Criterion for convergence of sequences

I know that $\{z\}=\bigcap\{\operatorname{cl}\,\{x_n\mid n\in S\} \mid S\subseteq \mathbb{N}\ \text{and}\ S\ \text{is infinite}\}$ is one of the criteria's of convergence of sequences in a metric ...
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### 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 ...
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### Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
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### Convergence criteria for interior

My book gives three equivalent statements as a theorem, under the heading "Convergence criteria for interior" : Suppose X is a metric space, z is in X and S is a non empty subset of X. First,z is in ...
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### Why is it that in a discrete metric space only eventually constant sequences are convergent?

I just read this result and was wondering what is the intuitive idea behind this ?
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### 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 ...
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Please consider the following question (note that $C_b$ is the space of bounded continuous functions): Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$. Prove $\{f_k \mid k = ... 1answer 351 views ### Convergence of a constant sequence and an eventually constant sequence I have to prove that a constant sequence and an eventually cinstant sequence is always convergent. I tried to do this as follows : I considered (xn) in X to be a constant sequence such that (xn) = ... 3answers 84 views ### Convergence of the sequence,$\frac 1n$Why does the sequence$\frac 1n$, where$n$is a natural number , does not converge when R is endowed with the discrete metric ? 1answer 89 views ### Question on a corollary of the Arzela-Ascoli theorem I am given a corollary of the Arzela-Ascoli theorem, and I've substantially rephrased it to this: If$S$is an equicontinuous and pointwise bounded set of functions with domain a compact metric ... 1answer 57 views ### Closure criterion for the convergence of sequences in a metric space Suppose$X$is a metric space,$z\in X$, and$(x_n)$is a sequence in$X$. Then according to the closure criterion for convergence of$(x_n)$in$X$we have that, $$\{z\} = ... 1answer 44 views ### Is the strong convergence of Borel probability measure metrizable? In a metric space (X,e), a sequence of Borel probability measure converges strongly, \mu_i \xrightarrow{s} \mu, iff for each Borel subset S \in X, we have \lim_{i \to \infty}\mu_i(S) = \mu(S). ... 1answer 80 views ### A question on Cauchy sub-sequences in a metric space (X,d) Let (X,d) be a metric space, and let (x_n) be a sequence in X. Prove that if (x_n) has a Cauchy subsequence, then for any decreasing sequence of positive \epsilon_k \rightarrow 0, there is a ... 1answer 65 views ### Another question about showing that a point is not an accumulation point of a given set Let C = \{ (\frac 1n , \frac mn) \in \mathbb{R}^2 : m,n \in \mathbb Z , n \neq 0 \} . I'm trying to argue that each point not on the y-axis is not an accumulation point of C. Is this ... 1answer 92 views ### A new kind of convergence Let (X,d) be a connected metric space. We say a sequence \{x_n\}_n \subseteq X is T-convergent to x \in X if the following is true:$$ \mbox{if} \; a,b \in X \mbox{ and } d(a,x) < d(x,b) ... 3answers 265 views ### If a fixed point is a limit of a subsequence of iterates, must the whole sequence converge to it? Say$X$is a compact metric space, with$f:X \to X$continuous. Now, for any$x_0$,$f^n(x_0)$must have a convergent subsequence, say$f^{n_i}(x_0) \to x_\infty$. If we know that any such limit is a ... 1answer 64 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 ... 1answer 79 views ### Prove that$d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$is a complete metric on$\mathbb{R}^k$. I am trying to prove that$d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$is a complete metric on$\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ... 2answers 550 views ### Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions Let$X, \ (Y, d)$be metric spaces,$f_1, f_2, \ldots \ : X \rightarrow Y$be continuous functions,$f: X \rightarrow Y$an arbitrary function. Prove that the following condtions are equivalent: 1) ... 2answers 70 views ### Find a convergent function in metric space Let$C[−1, 1]$be the space of continuous functions equipped with the metric$p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions$(f_n):[−1,1]\rightarrow \mathbb{R}$... 0answers 68 views ### convergence in metric space Let$C[-1, 1]$be the space of continuous functions equipped with the metric$(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence$(f_n)$of functions$f_n : [-1, 1] \to ...
Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in X$ with respect to metric $d_1$. ...
### 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 ...