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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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Pointwise convergence of $\sum_{n=0}^\infty \frac{1}{2^n\sqrt{1+nx}}$

Given this series : $$ \sum_{n=0}^\infty \frac{1}{2^n*\sqrt{1+nx}} $$ I have to prove for which $x \geq 0$ the series converges pointwise. if $x=0$ the series is : $$\sum_{n=0}^\infty \Big(\frac{1}{...
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1answer
47 views

Uniform Convergence of a subsequence over an arbitrary interval

I am getting ready for an entrance exam that is in August and I am trying to get a head start on the analysis section. I recently came across this problem and it is giving me some trouble. Consider ...
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1answer
12 views

With what extra condition interchange limit and integration allowed?

$f_n$ is sequence of continuous function which converges pointwise to continuous function. with what condition we can interchange limit and integration? I know that if sequence converges uniformly ...
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1answer
18 views

Proving a sequence of functions converges, is differentiable, etc.

I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative. ...
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18 views

Pointwise and uniform convergence $\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$

I am in deadlock studyibg the pointwise and uniform convergence of the following series: $$\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$$ Maybe should I handle it as a power series? But how? Any ...
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3answers
35 views

Pointwise limit function of a piecewise function

Find the pointwise limit function of: $$f_n(x)=\begin{cases} 0 & |x|> 1/n \\ nx+1 & x \in [-1/n, 0) \\ 1-nx & x \in [0, 1/n] \end{cases} $$ I think that in ...
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1answer
19 views

Bounded sequence of functions implies convergent subsequence

Here you can see my attempt at the proof. I am sure I did something wrong because my prof asked me to show it for rationals and I "somehow" showed it for all reals. I would appreciate it if someone ...
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5answers
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What is the main difference between pointwise and uniform convergence as defined here?

I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them. Let $E$ be a non-empty subset of $\Bbb{R}$. A sequence of functions $\...
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1answer
58 views

Suppose that $f_n\to f$ and $g_n\to g$, as $n\to \infty,$ uniformly. Then, $f_n g_n\to fg,$ as $n\to \infty,$ pointwise on $E$.

Can you, please, check if the following proof is correct? Thanks for your time and effort. Suppose that $f_n\to f$ and $g_n\to g$, as $n\to \infty,$ uniformly on $E\subseteq \Bbb{R}.$ Then, $f_n ...
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In functional analysis, is there a commonly accepted short-hand notation for specific types of convergence?

In math literature on functional analysis I found various short-hand notations for specific types of convergence, e.g. a single right arrow for pointwise convergence $$f_n(x) \underset{n \to \infty}{\...
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1answer
28 views

Manipulations with convergence a.e.

Let functions $f_n$ be measurable, $n \in N$, $f_n\rightarrow f$ almost everywhere. Prove that $\operatorname{arctg}f_n \rightarrow \operatorname{arctg}f$ almost everywhere. Honestly speaking, we ...
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1answer
29 views

Upper semicontinuous function as a poinwise limit of continuous fuctions

The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous ...
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Question on Existence of Limit from Monotone Convergence Theorem for non-negative measurable functions

Note $E^{*}$ is the space of non-negative measurable functions. In our lectures, it is written: Let $(f_{n})_{n}\subseteq E^{*}$ and $0 \leq f_{n} \leq f_{n+1}$, $\forall n \in \mathbb N$. $\...
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1answer
35 views

Do the property of $f_n$ carry over to $f_n'$?

Suppose a sequence of differentiable functions $f _ { n } : \mathbb { R } \rightarrow [ 0,1 ]$ converges pointwise to the zero function. Does it follow that the derivatives $f _ { n } ^ { \prime }$ ...
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1answer
32 views

Proof that pointwise convergence can disrupt convergence

I'm trying to get a grasp on point-wise convergence and am hoping to prove something to give a concrete example of why it's weak. The lemma goes as follows . Suppose $f _ { n } : [ a , b ] \...
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1answer
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Examples of some Pointwise Convergent Sequences of Functions

I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...
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2answers
40 views

Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
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2answers
51 views

Prove that the function $f_{n}(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$ converges pointwise for $x\in \mathbb{R}$.

I want to show that the function $$f(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}} $$ converges pointwise for all $x\in \mathbb{R}$. Furthermore, there are some intervals $(a, b)$ on which the function ...
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$f_n(x) = n^2x^n(1-x)^2 $ uniformly convergent? [closed]

Let $$f_n(x) = n^2x^n(1-x)^2\quad \text{on $I = [0,1]$}$$ Find the pointwise limit and determine whether or not the convergence is uniform and provide reasoning. Hint: consider $f_n=(1-n^{-1})$. ...
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2answers
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Prove $f_n \to f$ uniformly on $\mathbb{R}$

Denote by $D$ the set of all continuous, increasing functions $f: \mathbb{R} \to [0, \infty)$ such that $\lim \limits_{x \to - \infty} f(x) = 0 $ and $\lim \limits_{x \to + \infty} f(x) = 1 $. If $f_n,...
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Ways of checking pointwise convergence

According to the definition of pointwise convergence: A sequence $f_m(x)$ of real valued functions defined on D a subset of real numbers is said to be pointwise convergent to$f(d)$ at a point $d\...
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2answers
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converge pointwise but not uniformly

How can I prove that $$\sum_{n=1}^\infty \frac{\sin(nx)}{\sqrt{n}}$$ converges pointwise on $[-\pi, \pi]$ but not uniformly? For the pointwise part, I tried to prove it by comparison, using $$\sum_{...
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1answer
37 views

Pointwise limit of the sequence of continuously differentiable functions defined inductively.

Let $f_1 : [-1, 1] \rightarrow \mathbb{R};\: f_1(0) = 0 $ be a continuously differentiable function and $\lambda > 1$. Consider the sequence of functions defined inductively by $f_k(x) := \lambda ...
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1answer
30 views

Determine whether the convergence is uniform or almost uniform

Let $f_n(x) = \frac{nx}{n^2x^2 + 1}$. For each domain X, given below, how do you determine whether this sequence converges pointwise. Then if it does is it possible find the limit function and ...
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2answers
49 views

Preserving Bijectivity under pointwise limit

The following question I have tried to solve and have also tried to find reference for. A reference would be appreciated. Suppose that $f_i, g_i :\mathbb{R^n} \to \mathbb{R^n}$ are continuous ...
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1answer
12 views

Convergence of complex sequence checkup

I have to find the region of convergence and analyze the the pointwise and uniform convergence of the following sequence: $$\psi_n(z)=\frac{e^{-inz^2}}{(n+1)\sqrt{n}}$$ What I did: What I did first ...
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1answer
28 views

$n\chi_{[0, 1/n]}$ converges pointwise

I am confused why the function $ f_n = n\chi_{[0, 1/n]}$ converges pointwise. As I remember from my first analysis course, when we prove the pointwise convergence, we should fix $x$ first. Then, for ...
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1answer
27 views

a proof that the pointswise limits of lower semicontinuous (lsc) functions is lsc

I have a question regarding a proof regarding lower semicontinuous functions (the proof of the claim below). Definition: We call a function $f\colon \mathbb{R}^n\to\mathbb{R}\cup \{\infty\}$ lower ...
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1answer
68 views

Pointwise and uniform convergence of a piecewise sequence of functions on the closed, punctured disk, $\overline{D}\prime(0,1)$.

Consider the sequence of functions $$f_n(z) = \begin{cases} n, & \text{if $0<|z|\leq\frac{1}{n}$} \\ \frac{1}{z^4}, & \text{if $\frac{1}{n}<|z|\leq1$} \end{cases} $$ for $n\geq 1$, on ...
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1answer
53 views

Pointwise convergence of average of continuous functions

Suppose that $(f^n)$ is a sequence of continuous functions from a compact metric space $A$ into a convex compact subset of the reals $B$. Is there a subsequence, $(f^{n_k})$, such that the sequence $...
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1answer
30 views

Convergence of the following sequence of functions.

For $n \ge 1$, let $$g_n(x) = \sin^2 \left (x + \frac 1 n \right ), x \in [0,\infty)$$ and $$f_n(x) = \int_{0}^{x} g_n (t)\ \mathrm {dt}.$$ Then $(1)$ $\{f_n \}$ converges pointwise to a ...
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1answer
114 views

Uniform convergence of $\sum\limits_{n=1}^∞n^{-x}(e^{\frac{x}{n^2}}-1)$

Pointwise and uniform convergence of the following series of functions: $$\sum_{n=1}^{\infty} n^{-x}\left(e^{\frac{x}{n^2}}-1\right).$$ Now, the series of function converges pointwise as $x \in (-1,...
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2answers
28 views

Strongly convergent subsequence $+$ point-wise convergence $\Rightarrow$ strong convergence?

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ has a strongly convergent subsequence, say $f_{n_k}$. Also, assume $f_n\to ...
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Prove $f(x,y)$ defined on $\mathbb{R}^2$ is Lebesgue Measurable if $f$ is continuous in each variable separately.

This problem is from Ziemer's Modern Real Analysis and there is a suggestion on the page which tells us to approximate $f$ in the variable $x$ by piecewise-linear continuous functions $f_n$ such that $...
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1answer
41 views

If $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable $E$, $\int _E f_nd\mu \to \int _E f d\mu$

Let $f_n:X\to [0,\infty]$ be measurable functions such that $f_n\to f$ pointwise. Then prove that if $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable sets $E$ we have $$\int _E f_nd\...
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1answer
39 views

$\mu(\bigcup_{n=N}^\infty\omega:|g_n(\omega)-g(\omega)|>\varepsilon)<\varepsilon$ then $g_n\to g$ pointwise a.e.

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $g_n : \Omega \to \bar{\mathbb{R}}, n \in \mathbb{N}$ and $g : \Omega \to \bar{\mathbb{R}}$ be measurable functions. Prove that if $$\forall \...
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1answer
93 views

Pointwise limit of continuous functions is continuous on a dense set

I'm stuck in understanding the proof of the following theorem given during a course: Let $X$ be a Baire space, and $(Y,d)$ a metric space. Let $f_n:X\to Y$ be a sequence of continuous function, ...
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2answers
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If $f_n(s) \rightarrow f(s)$ for all s. Is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$?

If $f_n(s) \rightarrow f(s)$ for all $s$, is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$? Are minimums of $f_n$ converging to minimum of $f$?
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1answer
26 views

Can we interchange integral and limit if the sequence of function is uniformly integrable?

Assumptions: 1) $h(x,\omega)$ is defined on $\forall\omega\in\Omega, \forall x\in R\setminus\{x_0\}$ where $\omega$ is a random variable. 2) Let $h(x_0,\omega)=\lim\limits_{x\rightarrow x_0}h(x,\...
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1answer
62 views

Why $L^2$ convergence, with Riemann Integral, does not imply pointwise convergence at any point?

Let $L^2[0,1]$ be the space of continuous square integrable functions, where we use the Riemann Integral, no Lebesgue allowed. Let $(f_n)_n$ be a sequence of continuous functions on $[0,1]$ and $f$ a ...
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Showing $\sqrt[4]{n}(Y_n/\sqrt{n} - 1) \xrightarrow{D} N(0,1)$ using characteristic functions, for $Y_n \sim Bin(n, 1/\sqrt n)$

Let $Y_n \sim Bin(n, 1/\sqrt n)$. Show that \begin{equation} \sqrt[4]{n}(Y_n/\sqrt{n} - 1) \xrightarrow{D} N(0,1) \end{equation} using that \begin{equation} X_n \xrightarrow{D} X \quad \text{ if ...
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1answer
23 views

Pointwise convergence and convergence of integral

Let $\Omega \subset \mathbb R^n, n \in \mathbb N,$ be a bounded domain. Suppose $f_n \in L^1(\Omega)$ converge pointwise to a function $f: \Omega \to \mathbb R$ and $(\int_\Omega f_n)_{n \in \mathbb N}...
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1answer
98 views

What exactly is the contradiction in proving that $h_n(x)$ does not converge uniformly on any bounded interval?

I am currently going through this pdf https://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch9.pdf and in Exercise 9.2b, page 3, we have the following question. Prove that $h_n(x)$ does not converges ...
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19 views

Converge in norm and converge point wise are not equivalent.

I have known that for sequences of functions in $C[0,1]$, converge point wise is not equivalent with converge in norm for that two special norms: Integral and Suprem. However, when it comes to any ...
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1answer
69 views

Does $\sum^{\infty}_{n=1}xe^{-nx}$ converge uniformly on $[0,\infty)$?

I was trying to solve this problem with a friend but we got stuck. TRIAL We claimed that the series does not converge uniformly which implies that it is not uniformly Cauchy, i.e., $\exists\,\...
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1answer
20 views

Interval of convergence, pointwise and absolute

Give the series $$\sum_{n=0}^{\infty} \dfrac{(x + 10)^n}{3^n (n+1)},$$ find the intervals which result in point-wise and absolute convergence. Applying the root test we have, $$L(x) = \lim\limits_{n ...
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0answers
58 views

Convergence of $\sum_{n=1}\frac{1}{1-x^n} $

Consider $$ f(x)=\sum_{n=1}^\infty\frac{1}{1-x^n} $$ I am trying to find all $x\in\mathbb{R}$ at which the above series converges, absolutely converges, and all intervals of $\mathbb{R}$ on which the ...
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1answer
26 views

Difference between convergent in Measure and convergent a.e.

I’m thinking about two problems: In $L^2([0,1],dx);$ If $f_n\rightarrow f$ In $L^2$, then $f_n\rightarrow f$ In measure. If $f_n\rightarrow f$ In $L^2$, then $f_n\rightarrow f$ Almost everywhere. ...
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3answers
68 views

Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

My analysis notes contains the following question: if $(f_n)_n$ is a sequence of functions of $A \subset \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R} \cup \{-\infty, +\infty\}$ an accumulation ...
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1answer
37 views

Why do we need to use dominated convergence theorem?

I was thinking of this problem: If $f_n\rightarrow f$ pointwise a.e. and $\lvert f_n\lvert \leq g$ for some $g\in L^p$, then prove that $f_n \rightarrow f$ in $L^p$. To use the dominated convergence ...