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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Sequence of monotonic functions converges pointwise to a monotone function [closed]

How to think about this . I can't proof of disproof this
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Almost uniform convergence on an open set, pointwise convergence on a closed set and an equality of a limit of an integral of the sequence to examine.

$f_n: [a,b] \rightarrow \mathbb{R}$ is a function series of continous functions almost uniformly convergent on $(a,b)$ and convergent pointwise on $[a,b]$. Examine if $$\lim_{n\rightarrow \infty}\...
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Sequence of Lipschitz maps that converge pointwise to a Lipscthiz functions must have bounded Lipschitz constants?

Let $f_n:M \to M$ be a sequence of Lipschitz maps in a metric space $(M,d)$. Assume that we know that $\{f_n\}_{n=1}^\infty$ converge pointwise to $f:M\to M$ that is also a Lipschitz map. Let $L_n$ ...
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2 answers
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Does uniform integrability imply almost sure convergence of a subsequence?

True or false? If $X_n$ is uniformly integrable then there exists a subsequence $X_{n_k}$ such that $X_{n_k}$ converges a.s. to a random variable $X$ as $k→\infty$. My attempt so far: Since $X_n$ is ...
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Pointwise convergence of specific sequence of functions

I am trying to prove that the following sequence of functions converges pointwise using $\delta-\epsilon$ notation: $$ f_n(x) = \begin{cases} 1 & \mathrm{if}\; \frac{1}{2} - \frac{1}{n} \leq x \...
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3 votes
1 answer
32 views

Convergence of integrals without the dominated convergence theorem

Let $t>0$ and $\{f_n\}_{n \geq 1} $ be a sequence of functions that do not converge pointwise to any integrable function, but where: \begin{equation} \int_0^t f_n(s) ds \rightarrow F(t) < \...
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1 answer
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Equality of infimal integral over family of functions, with and without pointwise closure

Question: Suppose that $(X, \mathcal{A})$ is a measurable space. Let $\pi$ be a probability measure which is supported on a finite subset of $X$. Let $\mathcal{F}$ be a family of functions $X \to (-\...
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1 answer
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Lebesgue Dominated Convergence Theorem, $\sin(\frac{x}{n})$ example

We have to evaluate the limit of a sequence $\{f_n\}$ which is $$\lim_{n\rightarrow \infty }\int \frac{n\sin(x/n)}{x(1+x^2)}\,dx$$ using Lebesgue Dominated Convergence Theorem. The hint is: $\int \...
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4 votes
1 answer
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problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
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1 answer
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$\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$

How can I show $\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$? It is $\frac{d}{dx}x^n=nx^{n-1}$ so I tried to solve it ...
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There exists such result for any convergence of sequence definiton

Assume that $L(x_n) $ stands for the limit of the sequence $(x_n) $ in some sense, not necessarily the usual limit. For example, could be the almost convergence limit or the statistical limit. I ...
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If {fn} converges pointwise to {f} and each {fn} is bounded in a closed interval [a,b], is {f} also bounded in [a,b]?

Is it possible to say that if $f_n$ converges pointwise to $f$ and each $f_n$ is bounded in the closed interval $[a,b]$, then $f$ is also bounded in $[a,b]$? As far as I know, boundness is not ...
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1 answer
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Let $f_n(x) = (x-\frac{1}{n})^2$ for $x\in[0,1]$. [duplicate]

Does the sequence $\{f_n\}$ converge pointwise in the set $[0,1]$? And the question asks to give the limit function as well. So far I have: $$f(x)=\lim_{n\rightarrow\infty}f_n(x)$$ $$f(x)=\lim_{n\...
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3 answers
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Show that sequence must be in $ℓ^\infty$

Given a sequence $(a_n) \in \mathbb{R}$ so that for every sequence $(x_n) \in c_0 : (a_nx_n) \in c_0$. Show that this implies that $(a_n)$ has to be in $ℓ^\infty$. My thoughts: $(a_nx_n) \in c_0$ ...
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1 vote
1 answer
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A topology product problem with sequence of continuous and unbounded functions

Consider $X=\{f:\mathbb{Q}\rightarrow \mathbb{R}\}$ (set of any functions), with the product topology. Is true that for any $f\in X$, there are $(f_n)_{n\in \mathbb{N}}$ sequence of continuous and ...
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2 votes
2 answers
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uniform convergence of $\sum_{n=1}^\infty (1-e^{-\frac xn})\sin(nx)$

I have this question that I'm struggling to do in my calc textbook Note that I have not been exposed to complex numbers yet The question is: let $a$ be a positive real number such that $0 < a < \...
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3 votes
1 answer
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A sort of converse of Banach-Steinhaus theorem.

$(X, \|•\|) $ and $(Y, \|•\|') $ be two normed space. $\begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)}: T \text{ is bounded } \}\end{align}$ $\|T\|_{op}=\sup\{\|Tx\|':\|x\|\le 1 \}$ ...
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2 votes
3 answers
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Why does the following sequence of functions converge pointwise?

Why does $\lim_{n\to ∞} \frac{1}{\sqrt x} \mathbb{1}_{[2^{-n-1},2^{-n}]}$ converge pointwise to $0$? I thought that the function isn't defined for $0$ when ${n\to ∞}$, because the set of the indicator ...
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$W^{1,\infty}$ convergence vs Lipschitz convergence

I have read that for a bounded open set $U$ with $C^1$ boundary, $C^{0,1}(U)=W^{1,\infty}(U)$. But is it also true that convergence in one norm implies convergence in the other? In particular, if $u_k\...
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1 vote
1 answer
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Uniform convergence of sequence of functions that converges pointwise to an unbounded function

I have the following sequence of functions before me: $f_n(x)=\dfrac{1-x^n}{1+x},-1<x<1$ I have to determine whether above sequence of functions is uniformly convergent or not. First of all, I ...
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2 votes
1 answer
60 views

Weak convergence and convergence of functionals in dual space

Let $X$ be a normed space, $(x_n)_{n \in \mathbb{N}} \subset X$ and $(x_n')_{n \in \mathbb{N}} \subset X'$ such that $(x_n)_{n \in \mathbb{N}}$ weakly converges to $x \in X$ and $(x_n')_{n \in \mathbb{...
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1 vote
1 answer
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Question about Convergence for sequence of recursively defined functions

Consider the sequence $(f_{n})_{n=1}^{\infty}$ of continuous functions on $I = [0, \infty)$ defined recursively by $f_{1}(x)=x, f_{n}(x)=x+\int_{0}^{x}f_{n-1}(t)\sin(x-t) dt, \forall n\geq 2$. This ...
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2 votes
1 answer
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Let $ f_n $ be a sequence of integrable functions on $ [0, +\infty ) $. Disprove the following statements regarding integrability.

Problem: Let $ f_n $ be a sequence of integrable functions on $ [0, +\infty ) $. Prove/Disprove the following statements: (A) If $ f_n $ converges pointwise to $ f(x) = 0 $ on $ [0, +\infty ) $ then $\...
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1 answer
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Proof of uniform convergence of converged equicontinuous sequence

Question: $\{f_n(x)\}$ is continuous on $[a,b]$ and has pointwise convergence on $[a,b]$ (to $f(x)$) . $\forall \epsilon>0,\exists\delta>0,\forall x,y\in[a,b](|x-y|<\delta),\forall n\geqslant ...
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$\displaystyle\sum_{n=1}^{\infty} \frac{x}{1+n^{\alpha}x^{\beta}}$ convergent iff $\alpha >1$, uniform convergent if $\beta\leq 1$ or $\alpha>\beta>1$

I am preparing for my exam and need help with the following tasks: Let a sequence be defined as $\displaystyle\sum_{n=1}^{\infty} \frac{x}{1+n^{\alpha}x^{\beta}}$ with $\alpha$ and $\beta$ be real ...
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3 votes
3 answers
102 views

Finding "limit-function" of $f_n=\frac{x}{1+n^2x^2}$, Pointwise/Uniform convergence of ${f_n'}$

I am preparing for my exam and need help with the following tasks: Let ${f_n}:[-1,1]\to \mathbb{R}$ be defined as $f_n=\frac{x}{1+n^2x^2}$ Is ${f_n}$ pointwise/uniformly convergent? Specify the "...
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1 vote
1 answer
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Pointwise and uniform convergence of ($f_{n}(x)=\frac{1}{e^{xn}}$)

I would like to know if the sequence $f_{n}(x)=\frac{1}{e^{xn}}$ converges pointwise and uniformly. $f_{n}:[0, \infty) \rightarrow \mathbb{R}$, for $n\in \mathbb{N}$. I think it is easier to show that ...
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-1 votes
1 answer
38 views

Construct a sequence of random variables

Let $a_n$ be a given sequence of numbers (constant), $X$ be a standard normal random variable. Construct a sequence of random variables $X_n$ such that $X_n \rightarrow X \text{ a.s.}$ and $E(X_n) = ...
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0 answers
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A topology $\tau$ on $ L([0,1])$ that induces the $almost-everywhere$ convergence on $[0,1]$

Let be $L([0,1])$ the vector space of equivalence classes of measurable function $f:[0,1]\rightarrow \mathbb{R}$ (I don't have more details about the type of equivalence that the exercise is talking ...
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1 answer
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Integral of a function by Lebesgue measure and monotony convergence

Let be $f:[0, 1] \rightarrow \mathbb{R}$ a function such that $\int_{[0,1]}|f|<+\infty$. Calculate: $$\lim_{n \rightarrow +\infty}\frac1n\int_0^1log(1+e^{nf(x)})dx$$ I wanted to either use the ...
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1 answer
60 views

Weak* convergence on $L^\infty(\Omega)$ and almost everywhere convergence [duplicate]

Let $\Omega$ be finite measure space. Suppose that $f_n \to f$ in $L^\infty(\Omega)$ for the weak* topology. Does there exists a subsequence (or a subnet) $(f_{n_k})$ such that $f_{n_k} \to f$ almost ...
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Find functions $f_k(x)$ that converge to $1$ point-wise, complying with conditions

I'm looking for continuous function $f_k(x), \forall x\in[0,1]$ which are $C^N$ for $x\in(0,1)$ with given $N>0$, complying with this conditions: $f_k(x)>0, \forall x\in(0,1)$ $f_k(0) = 0$ $...
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0 votes
1 answer
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How to prove that the convergence of the sequence $f_n(x) = \Big(1+\frac{1}{nx}\Big)^{nx}$ is not uniform.

Let $f_n(x) = \Big(1+\frac{1}{nx}\Big)^{nx}$ for $x\in (0, 1) $. Obviously the sequence converges pointwise and the limit function is: $$\lim_{n \to \infty}f_n(x) = \lim_{n \to \infty}\Big(1+\frac{1}...
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2 votes
0 answers
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Does $L^2$ convergence imply that the pointwise limit is the most frequent accumulation point?

The typical example for a function that converges in $L^2([0,1])$ but not pointwise is the indicator function with width $1/n$ that wanders accross, the interval, i.e. $$h_1 = 1_{[0, \frac{1}{2}]}, ...
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3 votes
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Divisible approximations to the uniform distribution

As a bunch of answers on this site have pointed out: Can sum of two random variables be uniformly distributed, https://mathoverflow.net/q/228014/5429, the Uniform distribution is not divisible. That ...
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1 vote
2 answers
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Uniform convergence and rigor proof of pointwise convergence of $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$ for all $x \in (0,\infty)$.

Let $(f_n)$ be a sequence of functions where $f_n:(0,\infty) \to \Bbb R$ defined by $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$. Show that $f_n$ does not converge uniformly on $(0,\infty)$. My ...
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2 votes
1 answer
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If $\mu(A_n\Delta A)\to0$, why is $A=\bigcup_{n\ge1}\bigcap_{m\ge n}A_m$? I get that it only holds for a subsequence

$\newcommand{\d}{\mathrm{d}}\newcommand{\M}{\mathcal{M}}\newcommand{\L}{\mathcal{L}^1}$Let $(X,\M,\mu)$ be a finite measure space. The following appears to be an unusual definition - I looked it up ...
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1 vote
1 answer
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convergence almost eveywhere of an uniformly bounded sequence of functions [closed]

Let $(\Omega,\mu)$ be a measure space. Suppose that $(f_n)$ converges almost everywhere to some function $f$ where each $f_n$ belongs to $L^\infty(\Omega)$. Suppose that $\sup_n \|f_n\|_\infty <\...
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3 votes
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necessary and sufficient conditions of $\frac{X_n}{a_n}\rightarrow0, \frac{S_n}{a_n}\rightarrow0$ and $\frac{max\{X_1,...,X_n\}}{a_n}\rightarrow0$.

$X_1,...,X_n$ are independent identical random variables of standard normal distribution, $S_n=X_1+...+X_n$, $a_n \uparrow \infty$, try to give the necessary and sufficient conditions of (1)$\frac{...
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1 vote
1 answer
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How can the pointwise and uniform limits of sequences of functions defined by parts be treated?

I don't think I have too much difficulty in understanding and showing that simple function converge pointwise or uniformly, the few "standard" examples, such as $f_n(x) = x^n$, $f_n(x)=\frac{...
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0 votes
1 answer
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Only pointwise convergence or also uniform convergence?

Does $f_{n}(x)=e^\frac{-x^2}{n}$, $x\in \mathbb{R}$ converge uniformly? It converges pointwise to $1$. For uniform convergence I need $|f_{n}(x)-f(x)|<\epsilon$. $\sup|f_{n}(x)-1|$ has to converge ...
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0 answers
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If the indicator of any closed(/open) set is the pointwise limit of continuous $0 \le f_n \le 1$, then so is the indicator of any constructible set?

FYI, a constructible set is one that can be generated by finite unions and intersections of closed and open sets, cf. my previous question. Question: Let $(X, \tau)$ be a topological space such that ...
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0 answers
24 views

Why does an increasing sequence of indices always exist in convergence in probability?

Consider the problem statement presented Convergence in probability with subsequences and its accepted solution by @honeybadger. Question: What I am struggling to understand is that why we can ...
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2 votes
0 answers
22 views

Show the sequence uniformly converge to null function. [duplicate]

Problem: If $f_0$ is a continuous function in $[0,a]$, $a>0$, show that the sequence $\{f_k\}$ (on that same interval) defined the recursive relatio n$f_k(x)=\displaystyle\int_0^x f_{k-1}(t)dt$ ...
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1 vote
0 answers
45 views

Limit under the sign of integral

I'm studying the following integral \begin{equation*} \int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau \end{equation*} in particular, its limit \begin{equation*} \lim_{a\to 0}\int_0^t \tau \cos(c+b\...
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0 votes
1 answer
22 views

Test pointwise and uniform convergence of the following sequences of functions on $[0,1]$: $f_{j}=x^{j}-x^{j+1}$ and $f_{j}=x^{j}-x^{2j}$.

Test pointwise and uniform convergence of the following sequences functions on $[0,1]$: $$f_{j}=x^{j}-x^{j+1}\;\;\text{ and }\;\;f_{j}=x^{j}-x^{2j}.$$ I've proved that both converge uniformly to $0$, ...
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2 votes
1 answer
44 views

Comparing different kinds of convergence

I want to gain a better understanding of the convergence of a sequence of functions $(u_n:I\to \mathbb R)_{n\in \mathbb N}$ in different norms, I know some results, for example that if $u_n(x)\to u(x)$...
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2 votes
1 answer
40 views

misconception about pointwise/uniform convergence

Can anyone clear up the misconception in my argumentation. If a sequence of functions $f_k(x)$ is pointwise convergent towards some limit $f(x)$ for every $x$, then at every point $x$ we can choose an ...
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0 votes
2 answers
64 views

Why pointwise convergence does not imply uniform convergence?

Definition 1. Suppose that $\left(f_{n}\right)$ is a sequence of functions $f_{n}: A \rightarrow \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Then $f_{n} \rightarrow f$ pointwise on $A$ if, for ...
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13 votes
2 answers
761 views

If $(f_n')$ converges uniformly on an interval, does $(f_n)$ converge?

Let $(f_n)$ be a sequence of functions that are all differentiable on an interval A, and suppose the sequence of derivatives $(f_n')$ converges uniformly on A to a limit function $g$. Does it follow ...
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