# 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.

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### Is the product of two uniformly convergent sequence of functions uniformly convergent?

I am doing the exercise V.1.7 of Amann and Escher's Analysis. In the exercise I am required to prove that the product of two uniformly convergent sequence of functions is uniformly convergent if just ...
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### A doubt on uniform limit of a sequence of uniformly continuous functions

I was looking at this question is MSE. Suppose $f_n:[0,1] \to \mathbb{R}$ is a sequence of uniformly continous functions and $f_n\to f$ uniformly. These are the questions that I put myself. What can ...
1 vote
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### If $f:[0,1]\to\mathbb R$ is continuous, then $f_n(x) = f(x^n)$ converges uniformly on $[0,a],$ $a < 1$ and $∫_0^1 f_n(x)\,dx \to f(0).$

Given f continues function in [0,1] we define a sequence of functions fn(x) = f(x^n) , given that for every number 0<a<1 the sequence of function is uniformly convergs in [0,a] to f(0) prove ...
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### Construct a non-increasing sequence (An) such that it converges to its supremum. [closed]

We know that a real sequence which is increasing and bounded above converges to its supremum. Is this possible for a non - increasing sequence as well? State that sequence. I am not able to think of a ...
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### Why is $C([0;1])$ with the supremum metric complete, but it is not with the integral metric

I am currently studying metric spaces in my mathematical analysis course and I came across two examples: First - show that the set of continuous functions on a closed interval (denoted as $C([a;b]$) ...
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### Has the composition of a sequence of quasiconformal mappings with unbounded dilatation and another q.c. mapping still unbounded dilatation?

Let $G \subseteq \mathbb{C}$ be a bounded domain in $\mathbb{C}$. Consider for $m \in \mathbb{N}$ a sequence of quasiconformal mappings $f_m: G \longrightarrow \mathbb{C}$ with unbounded maximal ...
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### Uniform convergence of the given series in the domain $\Bbb R$ [duplicate]

Check whether the following series of function is uniformly convergent in $\Bbb R$. $$\sum_{n=1}^\infty (-1)^n \frac{x^2+n}{n^2}.$$ If the ddomain is any bounded interval then by Weierstrass M-test ...
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### Pointwise convergence of a function where convergence to multiple values occur at a single point

Consider, $$f_n(x)=\frac{1-nx^2}{(1+nx^2)^2}$$ where, $$x \in \mathbb{R}, n \in \mathbb{N}$$ It is clear to me that not including $x = 0$, each function point converges to $0$ as $n \to \infty$. The ...
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### Differentiability of the limits of sequences of functions between Banach spaces

$\newcommand{\vertiii}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}$Let $(E, |\cdot|_E)$ be a real Banach space and $X$ an ...
1 vote
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### Differentiability of the limits of sequences of functions from $\mathbb R$ to a Banach space

Let $(E, \|\cdot\|)$ be a real Banach space and $X$ an open subset of $\mathbb R$. I'm trying to prove below result which is used in the proof of Hille-Yosida theorem in Brezis' Functional Analysis. ...
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### Can a convergent sequence have divergent subsequences? [duplicate]

I cannot seem to think of any example of a convergent sequence that has divergent sub-sequences. I am aware that a divergent sequence can have convergent sub-sequences. $(-1)^n$ is divergent and has ...
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### I wonder why the author assumed $(f_n)_{n\in\mathbb{N}}$ converges pointwise to a function $f$ on $[a,b]$. ("Calculus I" by Shizuo Miyazima)

I am reading "Calculus I" (in Japanese) by Shizuo Miyazima. Theorem 6.5 Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of continuous functions on $[a,b]$. Suppose $(f_n)_{n\in\mathbb{N}}$ ...
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1 vote