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For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

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Sequence of functions $(f_n)$ with $|f_{n+1}(x)-f_n(x)| \le \frac{1}{2^n+x^2}$ is uniformly convergent in $\Bbb R$

Let $(f_n)$ be sequence of functions with $f_n : \Bbb R \to \Bbb R$ satisfying $$|f_{n+1}(x)-f_n(x)| \le \frac{1}{2^n+x^2}, \quad \forall x \in \Bbb R, \forall n \in \Bbb N.$$ Show that $(f_n)$ is ... user avatar gerrr

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Do I get uniform convergence in $\mathbb R^N$ for this sequence of continuous functions?

Suppose I have a sequence of Lipschitz continuous functions (but not bounded) $z^k \colon \mathbb R^N \to \mathbb R, k \geq 1$ such that \begin{align} \sup_{\mathbb R^N} |z^{k+1}-z^{k} |\leq \frac {C} ... user avatar carlos85

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$L_\infty$ norm and uniform convergence with Lebesgue integral

Let $\Omega \subseteq \Bbb R^d$ be a bounded and measurable set and let $f, f_n : \Omega \to \Bbb R\cup \{±∞\}$ for $n ∈ \Bbb N$ be measurable such that : $f, f_n ∈ L^1(\Omega)$ for all $n ∈ N$ , $\... user avatar Kilkik

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How to make any number of 3 digits include 0-5 converge to base -31 (404)?

Firstly,I'm new to calculus.On Khan Academy,I have looked at divergence and convergence.I am wondering how on can converge to 404 or base -31. Secondly,I would love the steps to generate a series of ... user avatar prometheus

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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}\... user avatar patrini

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Prove that countable intersection/union of zero/cozero sets is a zero/cozero set.

Definition A subset $Z$ of a topological space $X$ is a zero-set if there exist a continuous real valued function $f:X\rightarrow\Bbb R$ such that $$ Z=f^{-1}[0] $$ So we say that a subset $Z^C$ of $X$... user avatar Antonio Maria Di Mauro

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$C^\infty$ of series of functions of two variables.

Let $a,b,c,d\in \mathbb R$, and $U:=[a,b]\times (c,d)\subset \mathbb R^2$. Suppose $f_n : U\to \mathbb R$ is $C^\infty$ for each $n\in \mathbb N.$ I'm considering why the statement below is true. If $\... user avatar SABAR

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Uniform convergence of $f_n(x)=\dfrac{x}{1+nx^2}$ using definition. What is wrong with my arguments? [duplicate]

Using $M_n$ test we get that above function is uniformly convergent on any closed interval. Hence, it is uniformly convergent on $[-1,1]$ as well. This implies that it is uniformly convergent on $(0,1)... user avatar Sohit Jatain

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Show the series $\sum\limits_{n=1}^{\infty} (\sqrt{1-x^{n}}-1)$ converges uniformly over the interval $[0,\frac{1}{2}]$. [closed]

So I have to prove that, the function series $\sum\limits_{n=1}^{\infty} (\sqrt{1-x^{n}}-1)$ converges uniformly over the interval $[0,\frac{1}{2}]$. How can I do that? user avatar SeriousLukas

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why is the Laplace transform of local integrable function with support on $[0,\infty)$ analytic?

There is a proposition about Laplace transform, but I don't know how to prove it. Let $f \in L^1_{loc}(\mathbb{R})$, $\operatorname{supp}(f) \subset[0, \infty)$, such that $a$ is the abscissa of ... user avatar xiuhua

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$f_n(x) =\frac{n}{2}\left(e^{\left(\frac{1}{n}-4\right)x}-e^{\left(-\frac{1}{n}-4\right)x}\right)$ - Prove uniform convergence in $[-1,1]$

$$f_n(x) =\frac{n}{2}\left(e^{\left(\frac{1}{n}-4\right)x}-e^{\left(-\frac{1}{n}-4\right)x}\right).$$ Need to prove uniform convergence in $[-1,1]$ I first calculated the P.W converge, which is: $e^{-... user avatar TryingToMath

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Where exactly does the series of functions $\sum_{n = 1}^{\infty} \frac{1}{n^2 (x+1)}$ converge uniformly?

For $x \in (-\infty; -2]\cup [0;+\infty): \frac{1}{n^2 (x+1)}\le \frac{1}{n^2} $, so it conv. uniformly on $(-\infty; -2]\cup [0;+\infty)$ by the Weierstrass M-Test. The problem is what happens when $... user avatar Sgg8

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Investigate whether the series $\sum_{n=1}^\infty \frac{1+x^2 \sin (5x)}{\sqrt{n^3}}$ converges uniformly on $[-4,2]$ or not.

Investigate whether or not the series $\sum_{n=1}^\infty \frac{1+x^2 \sin (5x)}{\sqrt{n^3}}$ converges uniformly on $[-4,2]$. Weierstrass M-Test. Let $(f_n)$ be a sequence of functions on $D \in \Bbb ... user avatar gerrr

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Show that if the series $\sum_{n=1}^\infty f_n$ is uniformly convergent on $\Bbb R$, then the sequence $(f_n)$ converges to $0$ on $\Bbb R$.

Let $f_n:\Bbb R \to \Bbb R$ be a function for which the series $\sum_{n=1}^\infty f_n$ is uniformly convergent on $\Bbb R$ for any $n \in \Bbb N$. Show that the sequence $(f_n)$ converges to $0$ on $\... user avatar gerrr

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Show that $f_n$ does not converge to $f$ in the uniform norm but does converge on the norm $||f||=\sqrt{\int_a^b|f(x)|^2}dx$

Bounded on $f_n:[0,1]\rightarrow\mathbb{R},$ $$ f_n(x)= \begin{cases} 1-nx&\text{if}\, 0\leq x \leq \frac{1}{n}\\ 1&\text{otherwise} \end{cases} $$ I think for the first part I need to ... user avatar John Warts

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