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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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
59 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
40 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
22 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 ...
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1answer
57 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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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
35 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
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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
19 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
39 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
18 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 ...
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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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1answer
26 views

If a function does not converge point-wise, then it doesn't converge uniformly as well, right?

Here is the function: $f_n(x) = \sqrt{\sin^2 x + n^{-4}}, \ \ x \in R$. If $x \neq \frac{\pi}{2}+2\pi k \text{ or } x \neq 2\pi k, \ \ k \in Z$ then the function doesn't converge at all even point ...
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Convergence of a series with bounds depending on the iteration

I have a positive sequence $f_k$. Every $T$ iterations, an event $A$ happens and I can prove that, for this iteration, I have the bound \begin{align} f_{k+1} - f_k \leq - (\alpha_k)^2 ~~~~~(1) \...
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1answer
71 views

Show that there doesn't exist a sequence of continuous functions $f_n$.

Show that there doesn't exist a sequence of continuous functions $f_n:\Bbb{R}\to\Bbb{R}$ that converges pointwise to $f$ such that $\forall x\in\Bbb{Q} \space f(x)=1$ and $\forall x\notin\Bbb{Q} \...
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1answer
28 views

Any $L^1$ function is the subtraction between two pointwise limits of continuous functions

Let $f\in L^1([0,1])$, and $\{\varphi_n\}_{n=1}^\infty$ and $\{\psi_n\}_{n=1}^\infty$ be two nondecreasing sequences of continuous functions on $[0,1]$. We want to prove that we can find two such ...
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2answers
38 views

set of pointwise convergence of series

Find the set of pointwise convergence of series: $$ \sum_{n=0}^{\infty} \frac{3^n}{2n^2+5}x^n $$ I think I need to use Cauchy-Hadamard theorem, but I don't know how to calculate this.
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1answer
31 views

Beta distribution converges in distribution to a binomial

I have the following problem. Let $X_n \sim \operatorname{Beta}(1/n,1/n)$ be a sequence of beta distributions, and $X \sim \operatorname{Bin}(1,1/2)$ which is equivalent to $\operatorname{bern}(1/2)$....
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1answer
21 views

Use Bisection Method for the function $f(x)=-1, x < 0, f(x) = 1, x \geq 1$ to find value of convergence

We are given the function $$ \begin{cases} f(x) = -1 & x < 0 \\ f(x) = 1 & x \geq 0 \end{cases} $$ Using the Bisection Method with starting values $a=-1, b=2$, we're asked ...
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0answers
35 views

Prove $∪_{k=1}^N[x_k − δ, x_k + δ] ⊃ [0, 1]$

Suppose ${f_n}$ is a sequence of nonnegative continuous functions defined on $[0, 1]$ and suppose $lim_{n→∞} f_n(x) = 0$ pointwise on $[0, 1]$. Prove that, $∀\varepsilon> 0$, there exist $δ ...
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0answers
40 views

Prove that $∪_{k=1}^N[x_k − δ, x_k + δ] ⊃ [0, 1]$

In the following question, Suppose ${fn}$ is a sequence of nonnegative continuous functions defined on [0, 1] and suppose $lim_{n→∞} f_n(x) = 0$ pointwise on $[0, 1]$. Prove that, $∀\...
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1answer
38 views

Show a step function is measurable

Suppose $X_n$ is a step function that converges pointwise to $X$, where $X: \Omega \rightarrow \mathbb{R} $ is a measurable function. How would I show that that $X_n$ is measurable with respect to $\...
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1answer
42 views

the proof of the topology of pointwise convergence

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, and let $Y^X : = \{f| f: X\to Y\}$ be the space of all functions from $X$ to $Y$. If $x_0 \in X$ and $V$ is an open set in $Y$, let $V^{(x_0)} \subseteq ...
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1answer
22 views

Showing this sequence in the sequence space converges to the zero sequence

Let $x^{(n)}$ be a the sequence in $\mathscr{l^2}(\mathbb{R})$ defined by $x_k^{(n)}:=\frac{1}{n+k}$. Show that the sequence $x^{(n)}$ converges to zero in $\mathscr{l^2}(\mathbb{R})$. Attempt: I ...
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0answers
69 views

Proof for problem similar to Dini's theorem

I am having trouble coming up with a proof for this problem about a sequence of continuous functions that converges (pointwise) to 0 on [0,1]. https://i.stack.imgur.com/FE7Wr.png Proving the first ...
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1answer
78 views

If $f_n\to f$, $f_n(x_n) = x_n$ and $f(x)=x$, does $x_n\to x$?

Let $f_1,\dots,f_n,\dots :\mathbb{R}_+\to\mathbb{R}$ be non increasing continuous functions such that $f_k(0)\geq 0$ and $\lim_{x\to+\infty}f_k(x)<\infty$ for all $k$. Let $f$ that have the same ...
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3answers
67 views

Pointwise convergence of series of functions implies uniform convergence?

In my Real Analysis class, a question came up, but I can't come up with a proof or disproof. Here's the conjecture: If $\sum f_n \rightarrow f$ pointwise on $(a, b) \subseteq \mathbb{R}$ then $\sum ...
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1answer
26 views

Almost sure convergence of an algorithm

You designed an algorithm and you proved that this algorithm converges almost surely. Let's note $X_n$ the value of your algorithm at step $n$, which is a random variable, and $X$ the true converged ...
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1answer
31 views

What does $\sum_{m=1}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$ converge to?

What does the following sum converge to as $n\rightarrow\infty$? $$\sum_{m=2}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$$ I think it converges to $-\theta^2/2$, but I am not ...
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1answer
23 views

Prove point-wise and uniform convergence of function (Carothers 10.9.f)

I am trying to complete exercise 10.9.f in Carothers Real Analysis. The ask is to provide a formal proof that $nxe^{-nx}$ converges pointwise and determine if it uniformly converges (if not, find a ...
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0answers
41 views

Pointwise convergence and $L^{p}$ bounded implies weak convergence in $L^{p'}$.

I would just like to know if my working here is sound. I want to show that given a sequence of function $(u_{n})\in L^{p}(\Omega)$ (with $|\Omega|<\infty$) such that $u_{n}(x)\to u(x)$ for every $...
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1answer
43 views

Does pointwise convergence of continuous functions imply uniform convergence in some interval? [closed]

Let $f_n$ be sequence of continuous functions which converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|...
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1answer
93 views

Does pointwise converge imply uniform convergence in some interval?

I was wondering if $f_n$ converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|_{E'}$. Currently, with ...
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1answer
36 views

Characterizing the set of convergence of a martingale using the compensator

Let $M_n$ be a square integrable martingale with $M_0 = 0$ and let $M_n^2 = M_0^2 + X_n + A_n$ be the Doob decomposition of $M_n^2$ i.e. $A_n = \sum_{i=1}^nE(M_i^2 - M_{i-1}^2|\mathcal{F}_{i-1}) = \...
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2answers
44 views

Will we have uniform convergence in this case?

Assume that $f: [a,b] \rightarrow \mathbb{R}$ is continuously differentiable. Then we know that for each $t \in [a,b]$ we have that $$\frac{f(t+\Delta t)-f(t)}{\Delta t}-f'(t)$$ will converge to ...
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1answer
46 views

Limit of sequence of functions $g_n(x) = n - n^2x $

I found the following sequence in an answer of a different question, see https://math.stackexchange.com/a/1919759/579544 Consider the sequence of functions $g_n : [0,1] \to \mathbb{R}$ defined by ...
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3answers
47 views

An increasing sequence of non-negative functions in $\mathcal{L}_{1}(X, \mu, \mathbb{R})$ is $\mathcal L_1$-Cauchy

To avoid any ambiguity, I first present the related definitions: Let $(X, \mathcal{A}, \mu)$ be a complete, $\sigma$-finite measure space and $(E,|\cdot|)$ a Banach space. We say $f \in E^{X}$ is $\...
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1answer
32 views

pointwise convergence of sequence of function

I need some help with this exercise: Let $n \in N$ and $f_n:[0,1]\Rightarrow R$ with $f_n(x):=\frac{n^\alpha \cdot \ x}{1+n^2x^2}$, and $\alpha\ge0$ a) For which $\alpha$ is $f_n$ pointwise ...
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1answer
39 views

On convergence and Lebesgue Integration

Good evening, I have some problems with following task because I don't understand the context between the forms of convergence and $L$-integrability : Let $f_n$ be an $L$-integrable sequence on $[1,\...
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1answer
19 views

About the convergence of integrals of absolute functions

Let $(X, \mathcal{M}, \mu)$ be a measure space. Let $f \in \mathcal{L}^1(X)$ and $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $\mathcal{L}^1(X)$ such that $$\int_X f \,d\mu = \lim_{n \...
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2answers
45 views

Let $f(x)$ be the $2π$-periodic function defined by $ f(x)= \begin{cases} 1+x&\,x \in \mathbb [0,π)\\ -x-2&\, x \in \mathbb [-π,0)\\ \end{cases} $

Let $f(x)$ be the $2π$-periodic function defined by $ f(x)= \begin{cases} 1+x&\,x \in \mathbb [0,π)\\ -x-2&\, x \in \mathbb [-π,0)\\ \end{cases} $ Then the Fourier series of $f$ $(A)$ ...
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1answer
49 views

Holomorphic functions which converge pointwise but not almost uniformly

I'm looking for an example of a sequence of functions $f_n$ which are holomorphic in the open unit disk $D(0,1)$ and converge pointwise for every $z \in D(0,1)$, but are not almost uniformly ...
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1answer
58 views

Convergence in Lp implies almost sure convergence of a subsequence

Assume that $f_n$ is a convergent sequence in $L^2(\Omega, L^2(0, 1))$ with limit $f$. Is it true that there exists a subsequence $f_{n_k}$ converging almost surely, i.e. such that $$ f_{n_k}(\omega) \...
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1answer
26 views

Does convergence almost everywhere imply convergence in probability in continuous time?

Let $\{X_t\}_{t\geq 0}$ a continuous-time stochastic process on a probability space $(\Omega, \mathcal F, \mathbb P)$. Assume $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Is it then ...
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0answers
22 views

Convergence on locally compact groups with an additional condition

This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto ...

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