Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

2
votes
1answer
42 views

A weird inequality regarding integrals, limits, as well as sequence of functions.

Consider a sequence of continuous function $f_n:[a,b]\to \mathbb{R}$. Suppose there exist constants $\gamma>1$ and $\beta>0$ independent of $n,p$ such that $$\left(\int_a^b|f_n(x)|^{p\gamma}dx\...
2
votes
1answer
37 views

Uniform convergence of $\frac{y/(2N)}{\sin(y/(2N))}$ towards 1

I can't come up with a proof, why $f_N(y) := \frac{\frac{y}{2N}}{\sin\left(\frac{y}{2N}\right)}$ converges uniformly against $1$ for $y\in(0,\pi),\ N\to\infty$. I would be thankful for any advice.
2
votes
1answer
113 views

Uniform convergence of sequence of function $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ on the interval $(1,+ \infty)$

I need to prove that the sequence of functions $f_n(x)$ is uniformly convergent to $f$ on the interval $(1,+ \infty)$. I've already shown that $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ is ...
1
vote
1answer
37 views

Uniform convergence of $f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$

I need to study the uniform convergence of $$f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$$ on the interval $[2,+\infty)$ I've shown that on $[0,+\infty)$: at $x =0$ $f_n(0)=0 \xrightarrow{} 0$ at $x \neq 0$...
1
vote
1answer
89 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 ...
1
vote
1answer
73 views

Uniform convergence and boundedness for the sequence of functions $\{f_{n}\}$

For $n \geq 1$, let $f_{n}(x) = x e^{-nx^2}$, $x \in \Bbb{R}$; Then the sequence $\{f_{n}\}$ is Uniformly convergent on $\Bbb{R}$?. I did this $f_{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ for ...
1
vote
1answer
62 views

Issue with pointwise limit to the indicator function on $\mathbb{Q}$.

As discussed in the following question it's impossible to have a pointwise limit of continuous functions converge to $1_{\mathbb{Q}\cap[0,1]}$. That said, why doesn't the following sequence of ...
1
vote
1answer
39 views

Finding limit of a convergent sequence

Let $(X,\mathcal{A},\mu)$ be a measure space and let $f_n:X\to [0,+\infty]$ be a sequence of $\mathcal{A}$-measurable functions. For each $n\in \mathbb{N}$, we define an incresing sequence $(g_{n,k})$ ...
1
vote
1answer
45 views

One-sided limits functions and monotony(proposition)

This is the proposition: A monotone function $f:D\rightarrow\mathbb R$, where $D$ is included in $\mathbb R$, has one-sided limits in every accumulation point of $D$ (not necessarily in $D$. I ...
0
votes
1answer
23 views

Convergence of Sequences of Functions Evaluated at Sequences in Their Domain

Suppose (1) A sequence of continuous functions$f_n(x)$ converges to a continuous function $f(x)$ pointwise on some set $I$, and (2) a sequence $\{x_n\}$ converges to $x \in I$ (****) Is it true ...
0
votes
1answer
19 views

show that $T_{k}(z)$= $\frac{p_{k}(z)}{(1-z)^{k+1}}$ for all $k \geq 1$ with $T_{0}(z)=\frac{1}{1-z}$ and $T_{k+1}(z)=1+zT'_{k}(z)$ $k \geq 0$

if $(T_{k}(z))_{k \geq 0}$ is sequence defined by $T_{0}(z)=\frac{1}{1-z}$ and $T_{k+1}(z)=1+zT'_{k}(z)$, $k$ $\geq 0$ show that $T_{k}(z)=\frac{p_{k}(z)}{(1-z)^{k+1}}$ for all $k \geq 1$ with $p_{...
0
votes
1answer
34 views

Proving that a sequence of functions has a convergent subsequence

Let $X$ be a metric space and $(f_n)$ an equicontinuous sequence of functions from $X$ to $\mathbb{R}$. We suppose that $A=\{x\in X| \{f_n(x),n=1,2,...\} \text{ is bounded}\}$, is not empty, and that $...
0
votes
1answer
17 views

How to show a function is locally C^1 implies globally C1?

Actually there is a series problem like f(x)=sum(n=1 to ∞)[sin(nx^2)/1+n^3], the question was whether f(x) is C^1 or not. This question has already answered, but a big issue of mine is I can't find ...
0
votes
1answer
39 views

Sequence of continuous function converging pointwise to continuous function is equicontinuous?

I've proven the following "theorem": Let $I \subset \mathbb{R}$ be an interval, $(f_n: I \rightarrow \mathbb{R})_{n \in \mathbb{N}}$ be a family of continuous functions converging pointwise to a ...
3
votes
0answers
83 views

Arzelá-Ascoli Theorem precompact sets

Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and ...
2
votes
0answers
38 views

Prove that $C_p[-\pi,\pi]$ is complete, knowing that $C[-\pi,\pi]$ is indeed complete

Show that the space $$C_p[-\pi,\pi] = \{f \in C[-\pi,\pi] \; | \; f(-\pi)=f(\pi) \}$$ is complete knowing that the space $C[-\pi,\pi]$, i.e. continuous functions on $[-\pi,\pi]$, is complete. Here is ...
2
votes
0answers
43 views

Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.

Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$. A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows: For $k \in ...
2
votes
0answers
184 views

$C^\infty_0$ approximation of $L^\infty$

I have recently proven that $C_0^\infty (\Omega)$ is dense in $L^p (\Omega)$ for $1\leq p < \infty$. It is known that $C_0^\infty (\Omega)$ isnt dense in $L^\infty (\Omega)$. However there is a ...
2
votes
0answers
40 views

limit function $f$ of $\{F_n\}$. $F_n=nx^n(1-x^2) $

Find the limit function $f$ of $\{F_n\}$. $F_n=nx^n(1-x^2) $ My solution: For $x=\pm1, 0, F_n=0 \implies f(0)=f(1)=f(-1)=0$. For $|x|>0$. we have $$(1-x^2)\lim_{n\to\infty} nx^n=\infty\text{ , ...
2
votes
0answers
396 views

Convergence in measure iff every subsequence has a convergant subsequence

My task is to show that on a finite measure space $(X,A)$, if $f_n \to f$ in measure, then every subsequence of $\{f_n\}$ has a subsequence that converges to $f$ almost everywhere. I believe I was ...
1
vote
0answers
35 views

Sum of an odd recursive sequence

Let $a_0 = 1$ $a_1 = 1 - \frac{e}{2}$ $a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$. Find $\sum_{r=0}^{\infty}a_r$.
1
vote
0answers
23 views

Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
1
vote
0answers
36 views

Combining little-o notation

Suppose we have $$f_{m}(n) = \frac{1}{n}g_{m}(n) + o(\frac{1}{n})$$ where the little-o notation is uniform in the variable $m$ as $n \rightarrow \infty$. Under what conditions is $f(m,n) = o(\frac{1}{...
1
vote
0answers
23 views

Using ∞-BinFractions to define a topological space homeomorphic to the positive real numbers?

First the proposed theory: Here $0 \in \mathbb N$. Definition: A non-constant function $f: \mathbb N \to \mathbb N$ is called a ∞-BinFraction if it satisfies the following: $\tag 1 \forall n \; f(n+...
1
vote
0answers
18 views

Constructing a uniform convergence sequence

Let $f : \overline{\Omega} \subset \mathbb{R}^{N} \to \mathbb{R}$ be a $C^{2}(\overline{\Omega})$ function. Can we always construct a sequence $f_{n}$ such that $f_{n} \to f$ uniformly in $\overline{\...
1
vote
0answers
16 views

Prove that the sequence of derivative functions converges uniformly on every interval [-M,M].

The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$. We are asked to show that the sequence of derivatives, $f_n'(...
1
vote
0answers
42 views

Similar to Dini's Theorem, without the hypothesis of the continuity of $f_{n}$? Is true?

Let $(f_{n})_{n \in \mathbb{N}}$ be a sequence of non-decreasing function of $[0,1]$ to $[0,1]$. Assume that (1) For any $x \in [0,1]$, the pointwise limit exist and $f(x) = \lim_{n \to \infty}f_{...
1
vote
0answers
61 views

Is this entrance exam question correct regarding uniform convergence?

Q. Consider the sequence of real-valued functions $\{f_n\}$ defined by $$f_n(x)=\frac {1}{1+nx^2}.$$ Assuming the fact that $\{f_n\}$ converges uniformly to a function $f$ find out real numbers $x$ ...
1
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0answers
24 views

$F$ normal in $H(\Omega) \iff$ normal in $U$

Prove $F$ is normal in $H(\Omega) \iff$ for every $z\in H(\Omega)$ there is a neighborhood $U$ of $z$ such that $F$ is normal in $U$. The direct implication is immediate because $U\subset \Omega$. ...
0
votes
0answers
18 views

Differentiable limit of a (uniformly convergent) sequence of differentiable function (again, but not exactly !)

If one has a uniformly convergent sequence of differentiable functions $f_n$ (say from $\mathbb{R}$ to $\mathbb{R}$), we know that the limit $f$ is not always differentiable. Even if it is ...
0
votes
0answers
26 views

Uniform Continuity implies boundedness in sequence of functions

We want to show that, letting $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions on $[0,1]$ that's uniformly convergent, then there exists $M>0$ such that $\forall n\in\mathbb{N}$, $|f_n(x)|\leq M$...
0
votes
0answers
26 views

Bounding the maximum of a sequence of continuous functions using integrals

I have absolutely non clue on how to solve this one. First, recall that $$\lim_{p \rightarrow +\infty }\left ({\int\limits_a^b |f(x)|^{p}dx)} \right) ^{\frac{1}{p}} = \max_{x \in [a,b]} |f(x)|$$ ...
0
votes
0answers
39 views

A function sequence that converges uniformly over $[ a, + \infty[ $ but doesn't for $ ]0,+\infty[$

I have a sequence function that I'm asked to check the point-wise convergence and uniform convergence: $f_n = ne^{-n^2x^2}$ for $x \in R$ The pointwise convergence is pretty easy to do, we find ...
0
votes
0answers
34 views

My possibly fraudulent proof of $f_n$ Cauchy in measure => $f_n -> liminf$ in measure

This result seems too convenient and I feel like Folland would have used this to prove proposition 2.30 if this was true. My "proof": Let $f_n$ be Cauchy in measure. By definition of $\liminf f_n(x)...
0
votes
0answers
30 views

Finding sequence of functions with compact support for Integral of given function

Let $a>0$ and $f(x):\mathbb{R}\rightarrow\mathbb{R}$ with $$f(x):\begin{cases}\frac{1}{\sqrt{a^2-x^2}},& x\in(-a,a)\\ 0,&\text{else} \end{cases}$$ I now have to construct sequences of ...
0
votes
0answers
30 views

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\...
0
votes
0answers
21 views

Prove that the sequence of derivative functions converges uniformly on every interval [-M,M].

The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$. We are asked to show that the sequence of derivatives, $f_n'(...
0
votes
0answers
25 views

Can i prove the limit of a function using the definition of sequence limits?

So I'm asked to prove the $\lim_{x\to 1} \frac{4x+3}{x^2+4x-3}$ and of course it can be easily found $\frac{7}{2}$. However, I know I can prove it with epsilon delta definition but frankly that's ...
0
votes
0answers
30 views

Understanding the integral of a sequence of functions

I have a integral with the following property: $$ \forall n>0,\quad \int_X f_n(x) dx> 0 $$ I am trying to be able to say that: $$ \int_X \hspace{.5em}\lim_{n\rightarrow 0} \hspace{.5em}f_n(x)dx ...
0
votes
0answers
15 views

Equicontinuity with respect to a sequence of points? (terminology question)

I'm searching for an established term for "equicontinuity with respect to a sequence of points" for a function. Let $\{f_k\}$ be a family of functions and $\{x_k\}$ some associated points. My version ...
0
votes
0answers
32 views

Under which hypothesis is a pointwise convergent sequence of functions, with a uniformly convergent subsequence, uniformly convergent?

I was trying to undestand the relation between pointwise and uniform convergence of sequences and subsequences of functions and the following question popped up: What conditions must be assumed over a ...
0
votes
0answers
28 views

How to prove that a sequence $f_{n} \in L^{1}$?

I'm trying to studying Banach and Hilbert spaces. I'm interested to clearly understand when a sequence of functions $f_{n}$ begin to a Banach space ($L^{1}$) or to a Hilbert space ($L^{2}$). I ...
0
votes
0answers
43 views

Convergence of sequence of functions to x^2

Given a function $g_n$ defined on $2^n$ subintervals of [0,2] such that for subinterval $[\frac{k-1}{2^{n-1}},\frac{k}{2^{n-1}}]$, $g_n(x) = (\frac{k-1}{2^{n-1}})^2.$ I am trying to show $g_n$ ...
0
votes
0answers
36 views

What can I say for this function sequence $n-n^2x$

I know it is easy but I am unable to see this function sequence’s limit. $f_n(x)=n-n^2x$ what can I say for limit of this sequence when $x$ in $(0,1/n)$ interval?
0
votes
0answers
71 views

Almost Uniform Convergence - enough to move limits into integrals? (check proof)

I have a specific function in mind, but I would like to know about the general theory. I have a sequence of functions $\lbrace W(x) \rbrace _{m\in\mathbb{N}}$. I know the following about $W$: at $W(...
0
votes
0answers
34 views

Find the limit function $f$ of $\{F_n\}$. $F_n=nx^n(1-x^2) $

Find the limit function $f$ of $\{F_n\}$. $F_n=nx^n(1-x^2) $ My solution: For $x=\pm1, 0, F_n=0 \implies f(0)=f(1)=f(-1)=0$. For $|x|>0$. we have $$(1-x^2)\lim_{n\to\infty} nx^n=\infty\text{ , ...
0
votes
0answers
28 views

Show that $f_n (x)$ converges to $f(x)=0 \ \forall x$

$f_n$ is defined on $[0,1] \ \ \forall n \in \mathbb{N} $, $f_n(x)=\begin{cases} n(1-nx) &\mbox{if } \ 0 \leq x < \frac{1}{n} \\ 0 & \mbox{if } \ \frac{1}{n} \leq x \leq 1 \end{cases} $ ...
0
votes
0answers
67 views

Example of a sequence $(f_n)_n$, such that $(f_ n')_n$ converges uniformly, but $(f_n)_n$ doesn't converge in any point.

I've seen the following theorem: If $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n(c))_n$ converges to $f(c)$ for some $c\in [a,b]$, and $(f_n')_n$ converges uniformly to $g$...
0
votes
0answers
101 views

Convergence of an uncountable sequence of functions

I have a function $g(\cdot)$ such that such that $\lim\limits_{t \to \infty}g(t-s)=0$. I need to show that $\lim\limits_{T \to \infty}\int_{-\infty}^tg(T-s)ds=0$, where $0<t\ll T$. I thought that ...
0
votes
0answers
57 views

On Uniform Convergence of Sequence and Series Of functions.

Let $f_n(x)=\displaystyle \frac {x^n}{1+x^2}$, for $n\in \Bbb N$. Then which of the following statements is true? (A)$\;\;f_n$ converges uniformly on $[0,1]$. (B)$\;\;f_n$ converges uniformly on $[-...