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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.

1
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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 $x$ ...
3
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3answers
75 views

Calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$.

I need calculate $\displaystyle\lim_{n\to\infty}\int_0^1\frac{nx}{1+n^2 x^3}\,dx$. I showed that $f_n(x)=\frac{nx}{1+n^2 x^3}$ converges pointwise to $f\equiv 0$ but does not converge uniformly in [0,...
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 $U\subset \Omega$. ...
1
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1answer
93 views

Relation between uniform convergence and derivatives

I know the following relation between uniform convergence and derivatives: If $(f_n)$ is a sequence of differentiable functions on $[a,b]$ such that $\lim_{n\to\infty} f_n(x_0)$ exists (and is ...
2
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0answers
391 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 ...
2
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2answers
118 views

Let $X$ be a metric space with metric $d$.Then $d:X \times X\rightarrow \mathbb R$ is continuous?

Let $X$ be a metric space with metric $d$.Then $d:X \times X\rightarrow \mathbb R$ is continuous? Please Check the following proof- We'll try to show this via sequential crieria of continuity....
0
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0answers
27 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
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0answers
40 views

Continuity of limit on a compact set implies uniform convergence [duplicate]

Suppose $f_n$ is a sequence of increasing functions defined on $[a,b] \to \mathbb{R}$. Say that $f_n \to f$ pointwise, such that $f$ is continuous. Does this imply that the convergence is also ...
2
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1answer
47 views

Suppose $f_n : [0,1]\rightarrow\mathbb{R}$ is a sequence of $C^1$ functions that converges pointwise to $f$.

I came across this problem in Davidson's real analysis text and wanted some input on whether my thinking is valid. This is section 8.1, problem H. Here is the full statement of the problem. Suppose ...
3
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2answers
64 views

Sequence of uniform convergent sequence $\{f_n\}$ such that $\{f_n'\}$ does not converge [closed]

I would like to find uniformly convergent sequence of differentiable functions $f_n:[0,1]\to\mathbb{R}$ such that the sequence $f_1'$, $f_2'$,$f_3'\ldots$ does not converge.
0
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1answer
246 views

Prove that the limit of a sequence of polynomials a polynomial

Suppose that $(f_n)$ is a convergent sequence of polynomials on D. prove that the limit function f is also a polynomial; or find a counterexample. Is this equivalent to "the limit of a sequence of ...
1
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2answers
945 views

Sequence of Measurable Functions Converging Pointwise

Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise almost every on $E$ to the function $f$. Then $f$ is measurable. Proof: Let $g : E \to \Bbb{R}$ be defined by $g(...
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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
1answer
452 views

Interchanging limit and supremum

Sequence of functions is bounded $f_k(x) \le c\ \forall\ k, \forall\ x \ge 0,\ c \in \mathbb{R}$ and decreasing $\forall\ x \ge 0$. Is it possible to show such inequality $$\lim_{x \to \infty} \sup_{k ...
-1
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1answer
17 views

Uniform and absolutely convergence of functions [closed]

Is there any relationship between absolutely convergence and uniform convergence of function . I know a sequence of function which converge uniformly but not absolutely . Please provide example or ...
0
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1answer
507 views

Linear convergence with Newton method multiple root

I've a function $f \in C^{3}(\mathbb{R})$ with $f(\alpha)=f'(\alpha)=0$ and $f''(\alpha) \neq 0 $ and I'm trying to proove that the Newton's method has a linear convergence and that my sequence is ...
1
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1answer
58 views

Sequence $\{f_n\}$ of non-neg. meas. on [0,1] s.t $f_n \to 0$ a.e but if $[a,b]\subset[0,1]$ then $\lim_{n\to\infty}\int_{a}^{b}f_n(x)dx =(b-a)$

I need a sequence $\{f_n\}$ of non-negative measurable functions on [0,1] s.t $f_n \to 0$ a.e but for all $[a,b] \subset [0,1]$ we have that $$\lim_{n\to\infty}\int_{a}^{b}f_n(x)dx =(b-a)$$ I was ...
0
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2answers
55 views

What is the meaning and purpose of this theorem of inductive definitions?

I'm in my first analysis class, so naturally we're beginning with developing the real number system. As a part of the discussion of the natural numbers and induction, this theorem has come up in my ...
2
votes
3answers
93 views

Example of a sequence of integrable functions on $[0,1]$ s.t. $\lim_{n\to\infty}\int_0^1|f_n(x)|\,dx = 0$ but $f_n$ not converges to $0$ a.e? [closed]

I need an example of a sequence of integrable functions on $[0,1]$ s.t. $$\lim_{n\to\infty}\int_0^1 |f_n(x)|\,dx = 0$$ but $f_n$ does not converge to $0$ a.e. Anyone can provide an example with a ...
0
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3answers
99 views

Study the uniform convergence of $\cos^nx\sin^{2n}x$

Study the uniform convergence of $f_n(x)=\cos^nx\sin^{2n}x$ on $\mathbb{R}$ My attempt: Obviuosly $f_n(x)$ converges pointwise to $0,$ so I would like to proof uniform convergence to $0.$ I ...
2
votes
1answer
81 views

Limit of integrals of trigonometric functions

If we have a sequence of functions defined by $f_n(x) = \frac{\sin((x + \frac 1n)^2) - \sin((x-\frac 1n)^2}{\sin(\frac 1n)},$ how can we find $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) \, d\lambda$ ...
3
votes
1answer
228 views

$\epsilon$/3 - Proof for converging sequence of functions

I tried to prove the following statement and would like to ask if it's okay like that. $\underline{Statement}$: Let [B,$||.||_B$] be a complete normed linear space, $T_n :B \rightarrow B$ a ...
6
votes
2answers
945 views

Uniform Convergence of $(\sin x)^n$

Is the sequence of functions $(\sin(x))^n$ uniformly convergent in $[0,\pi]$? Can you give me a hint or solution, I have already already prove that it is UC in $[0,1]$ but I don't know how to proceed ...
0
votes
1answer
143 views

Check of uniform convergence of a sequence of function

The given sequence of functions is $$ f_n = -\frac{e^{-x^2n^2}}{n} $$ Prove that $f_n$ tends to $0$ uniformly; Prove that $f_n'$ tends to $0$ pointwise but not uniformly. I have tried ...
0
votes
1answer
55 views

Uniform convergence on $[1/2,1]$

Let $f_n(x)=\frac{1}{1+n^2x^2}$ for $n\in\mathbb{N},x\in\mathbb{R}$. Which of the following are true? $f_n$ converges pointwise on $[0,1]$ to a continuous function $f_n$ converges uniformly ...
0
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3answers
32 views

does the following function converge uniformly?

Does $$f_n(x)=\frac{x^n}{n^{\frac{1}{4}}}$$ converge uniformly on [0,1]? I know it converges pointwise.
2
votes
2answers
185 views

Sequence of differentiable,equicontinuous functions

I got stuck the other day trying to tackle the following problem : Let $ \left \{ f_n\left. \right \} \right. $ be a sequence of differentiable functions : $ f_n \quad : [0,1] \to \mathbb{R} $ ...
3
votes
2answers
99 views

On limit of point wise convergent sequence of continuous functions on real line

Let $\{f_n\}$ be a sequence of continuous functions on real line which is point wise convergent . Then is it true that for every $c\in \mathbb R$ , the set $\{x \in \mathbb R : \lim_{n\to \infty} f_n(...
3
votes
1answer
763 views

Given sequence of $L-$Lipschitz functions which converges pointwise, prove uniform convergence

Let $f_n:[a.b]\rightarrow \mathbb{R}$ be sequence of $L-$Lipschitz functions, that is: $$\forall x,y\in[a,b]: |f_n(x)-f_n(y)|\leq L|x-y|$$ Suppose $f_n \rightarrow f$ pointwise, prove $f_n \...
3
votes
1answer
81 views

Given $P_n(x)=\frac{n}{1+n^2x^2},$ Prove $f_n(x)=\frac{1}{\pi}\int_{-\infty}^\infty f(x-t)P_n(t)$ converges uniformly

Let $P_n:\mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions defined by $P_n(x)=\frac{n}{1+n^2x^2}$. $(a)$ Prove $\int_{-\infty}^\infty P_n(x) \mathop{dx} = \pi ,$ and that $$\forall \...
-1
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1answer
46 views

Constructing a sequence of function [closed]

Construct a sequence of functions on [0,1] each of which is discontinuous at every point on [0,1] and which converges uniformly to a function that is continuous at every point?
5
votes
4answers
556 views

Pointwise convergence to a uniform continuous function

What can we say about a sequence of functions that is pointwise convergent (over $R$) to a uniform continuous function? Does it converge uniformly? I have tried it using the definition but can't get ...
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
3answers
60 views

Does $f_n(x)=e^{\frac{-x}{n}}$ converge uniformly to $1$ on $[0,\infty)$?

Given $f_n:[0,\infty)\rightarrow \mathbb R$ with $$f_n(x)=e^{\frac{-x}{n}}, $$ it's easy to see that this sequence converges pointwise to $f(x)=1$. However I am sure that if convergence is uniform ...
4
votes
2answers
127 views

Infinite points in a line (Contest Math)

Today, the Central American and Caribbean Mathematical Olympiad was held in El Salvador. Problem 6 was as follows: Let $k$ be an integer greater than $1$. Initially, Tita the frog is sitting at ...
1
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1answer
57 views

Sequence of functions convergence investigation and question

So i have a sequence of functions defined as: $$f_n(x)= \frac{nx}{1+n^2x^2}$$ So i have to investigate convergence. So first i found limit function: $$F(x)=0$$ So that's for pointwise convergence....
1
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1answer
69 views

Let $\{f_n\}$ be a sequence of unif cont functions from $(X,d) \to (\Omega,\rho)$ s.t $\{f_n\} \to f$ uniformly. Show that $f$ is unif cont.

This is what I have so far: $\textbf{Proof:}$ We must show that $f$ is uniformly continuous, hence satisfy that $\forall \epsilon > 0, \exists \delta > 0, \forall x,y \in X$ with $d(x,y) < ...
2
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1answer
44 views

Pointwise and uniform convergence application

Let $(f_{n})_{n}$ be a sequence of functions $f_{n}: \mathbb{R} \to \mathbb{R} $. They converge pointwise to a function $f: \mathbb{R} \to \mathbb{R} $. We know that the convergence is uniform on ...
2
votes
4answers
83 views

Show that $\underset {x \in [0,1]} {\sup} f_n(x) \rightarrow \underset {x \in [0,1]} {\sup} f(x)$ as $n \rightarrow \infty$.

Let $\{f_n\}$ be a sequence of continuous functions converging uniformly to a function $f$ on $[0,1]$. Then show that $\sup\limits{x \in [0,1]} f_n(x) \rightarrow \sup\limits_{x\in[0,1]} f(x)$ as $n \...
0
votes
1answer
47 views

Show that $f_n$ converges uniformly on $[-\frac {1} {2} , \frac {1} {2}]$.

Let $\{a_n\}$ be a sequence of real numbers and let $f_n(x) = \sum_{i=0}^{n} a_i x^i$. Suppose $\lim_{n \rightarrow \infty} f_n(1)$ exists. Show that $f_n$ converges uniformly on $[-\frac {1} {2} , \...
2
votes
1answer
85 views

Show that the sequence of functions $\{f_n\}$ convereges uniformly to $f$ on $[0,1]$ by the given condition.

Let $f$ be a continuous function on $[0,1]$. Suppose $\{f_n\}$ is a sequence of continuous functions on $[0,1]$ such that for any sequence $\{x_n\}$ in $[0,1]$, if $x_n \rightarrow x$ then $f_n(x_n) \...
1
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1answer
43 views

Find sequence of function that has several conditions

$\require{cancel}$I have to find a sequence of function $f_n(x)$ that would have these conditions: $f_n(x)$ is continious in the range $[0;1]$ $f_n(x)\rightarrow 0$, $x\in[0,1]\; \forall{x}$ $\int_0^...
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 $[-...
0
votes
1answer
221 views

Uniform convergence of sequence of functions defined on $[0,1]$

Let $f_n (x) =\frac{ x}{1+ nx^2} $ where $x \in [0,1]$ Then, $\lim f_n(x) = 0$ as $n \to \infty$ and so that $\langle f_n (x) \rangle$ converges pointwise to function $f(x) = 0$ on $[0,1]$. ...
0
votes
3answers
56 views

Uniform convergence of the sequence of funtions for $x \in [a,\infty)$ with $a>0$

Show that $f_n= \frac {2n^2x}{(1+n^2 x^2) \ln(1+n)}$ does converges uniformly in $x\in [a,\infty)$ for $a>0$ How do I prove the uniform convergence for that interval? How differently is ...
0
votes
1answer
29 views

convergence of a complex sequence of function

$f_n(z)=\frac{z^n}{n}$, where it is uniformly convergent? well, $\frac{z^n}{n}\to 0\forall |z|\le 1$ am I right?
2
votes
1answer
299 views

Showing that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not converge uniformly

I am trying to show that $f_n(x) = \frac{nx}{1+n^2x^2}$ where $x\in \mathbb{R}$ does not converge uniformly. I have made an attempt and would like to make sure that I am going about it in the right ...
0
votes
1answer
161 views

Uniform convergence of $\lim_{n \to \infty} \frac{nx}{1 + n^2x^2} \text{ where } 0 \le x \le 1$ [duplicate]

Prepering to second exam in Calculus 2, and I remember this question from the first exam that I fell on: Let $f_n(x)$ be the function sequence $\{\frac{nx}{1 + n^2x^2}\}_{n=1}^\infty$ Let $f(...