Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

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45 views

Point-wise convergence of $f_{n,m}(x) = \cos^{2n}(m!)\pi x$

I have recently been trying some questions on convergence of sequence of functions.I got stuck in one of the problems in which I am supposed to find the point-wise limit and discuss the uniform ...
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4answers
27 views

Uniform convergence of sequence of functions $\frac{2+nx^2}{2+nx}$ on [0,1]?

I have recently been trying some questions related to the uniform convergence of a sequence of functions. And meanwhile, I got stuck in one of the problems in which I have been supposed to discuss the ...
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1answer
36 views

The difference between the statements for sequences of function $f_n(x)$

Let I be an interval and c ∈ I. Statement A: For all $\epsilon$ > 0, there is $\delta$ > 0 such that,for all $n ∈ \mathbb{N}$ and for all $x ∈ I$ satisfying $|x−c|≤\delta$, $|f_n(x)−f_n(c)| ≤ \...
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2answers
45 views

Show that $\sum_{1}^\infty\frac{\sin(nx)}{n^3}$ is differentiable everywhere

I have recently been trying out some questions on series of functions.In one of the questions, I was given a series $$\sum_{1}^\infty\frac{\sin(nx)}{n^3}$$ and now I am supposed to show that the above ...
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2answers
35 views

Uniform Convergence of a series of functions using the Dirichlet's test

I have recently been trying out some questions on series of functions. I got stuck in one of those problems in which I am supposed to show that the below series of functions is uniformly convergent on ...
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1answer
17 views

Point-wise convergence of function

I have been trying out some questions on sequence of functions.In one of those questions,I am supposed to find the point-wise limit of the following sequence of functions defined on [$0,1$] as $$f_n(x)...
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0answers
16 views

Convergence properties of solutions to a sequence of linear programming problems

Consider the following linear programming problem $\mathrm{LP}$: \begin{align*} \text{maximize } & \mathbf{c} \cdot \mathbf{x}\\ \text{subject to } & \mathbf{A} \cdot \mathbf{x} \leq \mathbf{...
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0answers
34 views

Limit of a function for $\epsilon>1$.

I have been trying some questions on the convergence of a sequence of functions and was wondering about an intermediate step in which we have $\epsilon\gt0$ and an $x$ such that $0<x<1$ and it ...
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1answer
35 views

Uniform convergence of $x^n$ using the definition

I have been trying to prove the uniform convergence of sequence of functions defined by $f_n(x)=x^n$ on $[0,k]$ where $k<1$ by the epsilon definition of uniform convergence. I have found the point-...
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2answers
37 views

Uniform Convergence of $\frac{n}{x+n}$

I was trying an exercise on uniform convergence of sequence of real-valued functions. I got stuck in a problem in which I am supposed to prove that sequence defined by $f_n(x)=\frac{(n)}{(x+n)}$ is ...
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2answers
21 views

Almost sure convergence of random variables with same mean and the difference goes to zero on the product

Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$. Suppose that for almost every couple of points $(\omega,\omega')$ in ...
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1answer
35 views

Is pointwise convergence permutation-invariant?

I am interested in convergence between countable sequences of real numbers. (Perhaps the definitions to follow are nonstandard. Sorry!) Say that the sequence $\langle \langle x^1_1,x^1_2,x^1_3,...\...
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1answer
31 views

Find $f(x)$ to which the given sequence of functions converges

$$f_{n}(x) = \begin{cases} \sin^{2}\pi x, & n≤ |x|≤n+1, \\ 0,& |x| < n \text{ or }|x|≥ n+1.\end{cases}$$ How can I find $f(x)$ to which $f_{n}(x)$ converges? I do always have problems ...
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1answer
92 views

Show that there exists a real number $R≥0$ such that, for all $x$, $y\in [0$, $1]$ and all $n \in \mathbb{N}$, $|g_n(x)−g_n(y)|\le R|x−y|$.

Assume that $(f_n)_n$ is a sequence of functions continuous on $[0$, $1]$, differentiable on $(0,1)$, that converge pointwise on that interval to a function $f$, and such that each $f_n$′ is bounded ...
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2answers
20 views

Proof for uniform convergence of sequence of functions

I was given this problem: These are my calculations and I'm asking for verification: Pointwise limit: $\lim_{n \to \infty} f_{n}(x) = \lim_{n \to \infty} \frac{x^{2n}}{1+x^{2n}} = \lim_{n \to \...
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0answers
28 views

A subsequence of $f^n$ converge pointwise where $f$ is analytic on the unit ball

This is a problem from my past Qual: "Let $\Omega$ be the unit disk and $f:\Omega\to \Omega$ be an analytic function. COnsider the sequence $\{f^n(z)\}$ where $f^n=f\circ\ldots\circ f$ ($n$--times). ...
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1answer
51 views

$f:D\to D$ is analytic then $f^{n_i}(z)$ converges pointwise for all $z$

This is a problem from my past Qual. "Let $D$ denote the unit disk and $f:D\to D$ be analytic. Show that there exists a sequence $n_i$ s.t. $f^{n_i}(z)$ converges pointwise for all $z\in D$. Here $f^...
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1answer
51 views

Prove that $\sum_{n=1}^\infty \frac{1}{n} \sin(\frac{x^2}{n})$ is pointwise convergent for $x \in \mathbb{R}$

For $x \in \mathbb{R}$ consider the series $$ S = \sum_{n=1}^\infty \frac{1}{n} \sin(\frac{x^2}{n}) $$ Then I have to prove that $S$ converges pointwise. My attempt: It follows from the mean value ...
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1answer
48 views

Sum converging to an integral - Riemann sum?

Let $\epsilon > 0$ and $L >1$ such that $\frac{L}{2\epsilon} \in \mathbb{N}$. Take $\Lambda_{\epsilon, L} :=\epsilon\mathbb{Z^{d}}/L\mathbb{Z}^{d}$. Suppose $f \in C^{1}(\mathbb{R}^{d}/L\mathbb{...
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2answers
121 views

Dominated Convergence Theorem and Holomorphic Functions

This is exercise 133Xc in Fremlin Volume 1: Let $(X,\Sigma,\mu)$ be a measure space and let $G\subset\mathbb{C}$ be open. Let $f:X\times G\to\mathbb{C}$ be a function and suppose that the derivative $...
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1answer
30 views

Convergence of sequence of rv in distribution and pointwise

Let $X \sim \mathcal{N}\left(\mu,\sigma^2\right)$ and define a sequence of random variables $$ M_n = \begin{cases} X, & n \text{ is even}\\ 3-X, & n \text{ is odd} \end{cases} $$ For which ...
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0answers
20 views

Does point wise convergence and convergence of ${L^p}$ norm imply ${L^p}$-convergence?

Let $(X_n)_{n \in \mathbb{N}}, X \in {L^p}(\mu)$ with $X_n \rightarrow X$ pointwise a.e. $||X_n||_{L^p} \rightarrow ||X||_{L^p}$ Does this already imply convergence in $L^p$?
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0answers
57 views

Why does $h\circ f_{n}\to h\circ f$ when $h$ is continuous and $f_n\to f$ pointwise?

Let $(X,d_{X})$ be a metric space, and for every integer $n\geq 1$, let $f_{n}:X\to\textbf{R}$ be a real-valued function. Suppose that $f_{n}$ converges pointwise to another function $f:X\to\textbf{R}$...
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2answers
53 views

Exercise involving pointwise and uniform convergence: why does it converge and why it does not converge?

(a) Let $(f_{n})_{n=1}^{\infty}$ be a sequence of functions from one metric space $(X,d_{X})$ to another $(Y,d_{Y})$, and let $f:X\to Y$ be another function from $X$ to $Y$. Show that if $f_{n}$ ...
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1answer
22 views

Pointwise convergence of a series $ \sum_{n=1}^\infty \frac{1}{\sqrt{n}}\left(e^{-\frac{x^2}{n}}-1 \right)) $

Consider the series for $x \in \mathbb{R}$ $$ \sum_{n=1}^\infty \frac{1}{\sqrt{n}}\left(e^{-\frac{x^2}{n}}-1 \right) $$ Then I have to prove that the series converges pointwise on $\mathbb{R}$. To ...
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1answer
30 views

Pointwise Convergence in Banach Space Implies Convergence in Operator Norm

Assume that $(a_n : V \rightarrow W, n \geq 0)$ is a sequence of continuous linear maps with $V$ is Banach space, $W$ a normed space such that $(a_n(v))_{n \geq 0}$ is convergent for any $v \leq V$. ...
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1answer
30 views

different type of convergence of random variable with summation

I have the following problem: Let $\xi_n: (\Omega, \mathcal F, \mathrm P) \to (\mathbb R^1, Bor)$ I need to find a connections between: 1. $\xi_n \to \xi$, such that $\sum_{n=1}^\infty|\xi_n - \xi| ...
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1answer
20 views

Pointwise/uniform/total convergence of a specific power series [closed]

When does this series converge pointwise/uniformly/totally? $$\sum_{n=1}^{+\infty}n{x^{n+1} \over (x+1)^n}$$ I'd show you my tries but I don't know what to do, help?
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3answers
29 views

Uniform Continuity of a Piecewise Function

Suppose you have the function $f : [0,1] \rightarrow \mathbb{R}$, with $f(x) = 0$ if $x \in [0,1)$ and $f(x)=1$ if $x=1$. Prove that it is uniformly continuous. I got this function as the pointwise ...
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0answers
28 views

Convergence of specific power series

I have to evaluate pointwise/uniform/total convergence of this series and I didn't quite understand how to do it. $$\sum_{k=2}^{+\infty}{\ln k \over 2+\sin k}x^k$$ For pointwise convergence: it ...
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0answers
38 views

Finding the Pointwise Limit of a Function

If I have a sequence of functions $f_n[0,2] \rightarrow \mathbb{R}$ where $f_n(x) = \frac{x^n}{2^n+n}$. If I attempt to find the pointwise limit, I work out that by taking $x \in [0,2]$: We can ...
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1answer
24 views

Show that this family of open sets forms a topology of pointwise convergence

This is Exercise 22.12 (b) on page 185 of Elementary Analysis, second edition, written by Kenneth Ross. I searched the site for similar questions and found a couple that looked similar (this and ...
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2answers
50 views

Pointwise convergence of $f_n(x+1/n)$

Assume that $f_n \to f$ pointwise (that is for all $x$ we have that $f_n(x) \to f(x)$). Let $x\in \mathbb{R}$. Is the following statement true? $$ \lim_{n \to \infty}f_n \left(x+\frac{1}{n}\right) = f(...
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1answer
23 views

Pointwise convergence of a sequence of functions $g_n $ on $(0,1]$

As stated in the title I am trying to prove $g_{n}=\sum_{k=1}^{2^n}\frac{2^{n}}{k}\chi_{\left(\left(\frac{k-1}{2^{n}}\right)^{2},\left(\frac{k}{2^{n}}\right)^{2}\right]}$ converges pointwise to $\frac{...
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1answer
56 views

True or False: convergence on L1 of a martingale Xn with E|Xn|=1

I have to prove whether the next statement is true or not: 'if {Xn} for n>=1 to infinitive it is such a martingale that for everything n>=1, Xn>=0 and E|Xn|=1, then the sequence {Xn} for n>=1 to ...
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1answer
28 views

Is the Weierstrass M-Test an equivalence?

Given $g_n(x)= \frac{nx}{1+n^2 x}$, $g_n : [0,1] \to \mathbb R, n \in \mathbb N, n\geq 1$, I am trying to show that $g_n$ converges uniformly. I have shown that it converges pointwise to $g:[0,1]\to\...
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1answer
38 views

Given $U\subseteq\mathbb{C}$ an open set and $f : X\rightarrow \mathbb{C}$ measurable then $f^{-1}(U)$ is measurable

Given the measurable space (X,$B$) where $B$ is a $\sigma$-algebra, I want to show that if $f : X\rightarrow \mathbb{C}$ is measurable (in the sense that, there exists a sequence of simple functions ...
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0answers
20 views

Almost sure convergence for GARCH(1,1)-process

I'm proving the conditions for strict stationarity of GARCH(1,1)-process: $$X_t=\sigma_t Z_t\qquad \sigma_t^2 = \alpha_0 + \alpha_1 X_{t-1}^2 + \beta_1\sigma_{t-1}^2.$$ We can rewrite the process to a ...
2
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1answer
48 views

Pointwise convergence of holomorphic functions on a dense set

Let $G$ be an open connected set and let $D \subset G$ be a dense set. Let $(f_n)$ be a sequence of holomorphic functions in $G$ and assume $f_n \rightarrow 0$ pointwisely on $D$. Can we deduce that $...
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0answers
50 views

Prove Uniform Convergence for $\{f_n\}$ [duplicate]

Suppose $\{f_n\}$ is an equicontinuous sequence of functions defined on $[0,1]$ and $\{f_n(r)\}$ converges $∀r ∈ \mathbb{Q} ∩ [0, 1]$. Prove that {$f_n$} converges uniformly on $[0, 1]$. There are ...
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1answer
44 views

Question of pointwise convergence of a sequence of functions

I have the following task: Define $f_n:[-1,1]\to \Bbb R$ by $$f_n(x)=\begin{cases}1 , \text{ for $-1 \leq x \leq -1/n$} \\ -\sin(n\pi x/2) , \text{ for $-1/n \leq x \leq 1/n$}\\-1 , \text{ for $1/n\...
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1answer
32 views

Riemann integral counterexample to dominated convergence theorem?

The dominated convergence theorem (and similar theorems) is often claimed to be what makes Lebesgue integration superior to Riemann integration. But we also have the result that any (positive) Riemann-...
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0answers
18 views

Mathematical intuition why the iterative Bellman update converges to the optimal solution

I know that the mathematical justification for using the Bellman-equation iteratively to find the optimal policy in Reinforcement Learning is based on convergence results. I wonder however whether ...
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1answer
12 views

point wise convergence and the indeterminate form - trivial question

I am just a beginner in Math and a little confused about the point-wise convergence. I a getting contradicting results between indeterminate form and point-wise convergence. Is it common? consider ...
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0answers
100 views

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
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1answer
18 views

Question about the non-uniform convergence of $f(x)=2nxe^{-nx^2}$

I came across an analysis problem where I am asked to determine if $f(x)=2nxe^{-nx^2}$ converges to zero on the interval $[0,1]$ a) point wise and b) uniformly. Part a was fine. I noted that $|f_n -f|...
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1answer
60 views

Prove that there exist infinitely many subsequences of $(f_m)_{m≥1}$ which converge at every point of $E$.

Let $E=\{1/n | n\in \mathbb {N}\}$. For each $m\in \mathbb {N}$ define $f_m: E\to \mathbb{R}$ by $$f_m(x)= \begin{cases} \cos (mx) & x\geq 1/m \\ 0 & 1/(m+10)\leq x < 1/m \\ ...
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1answer
38 views

Sequence of continuous fuctions with compact support converges to 1.

Let $X\subset\mathbb{R}^d$ be an unbounded closed set and $C_0$ is the space of all continuous functions $h: X\to \mathbb{R}$ with compact support. I'm searching for the sequence $\{ h_n \} \subset ...
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2answers
17 views

Evaluate point convergence and uniform convergence of given funcion sequence

I'm given a function sequence as such: $$ f_n(x) = \frac{x^n}{n^n}$$ over $$x \in [0,1]$$ I have to find what is the point convergence and uniform convergence of this sequence. What is the proper ...
2
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0answers
43 views

Weak and Pointwise Convergence question

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...

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