# Questions tagged [uniform-convergence]

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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### Question about applying Dominated Convergence Theorem

Question: $\phi_n(x)=\int_{x_0}^xf(t,\phi_n(t))dt$ ,where $\phi_n(x)$ is continuous on $(a,b)$ and $f$ is continuous and bounded on $(a,b)\times(-\infty,+\infty)$. 1.If $\phi_n(x)$ converges uniformly ...
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### Prove uniform convergence of the series $\frac{(-1)^n}{n+x}$ in $[0,\infty)$

Can it be proved without using dirichlet test for uniform convergence? Edit: Answer is Cauchy Criteria for Uniform Convergence
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### How to prove that this series of functions converges uniformly?

Let $z_0 \in \mathbb{C} \setminus \{0\}$ be a fixed complex number and let $z \in \mathbb{C}$. Let $K \subset \mathbb{C}$ be a compact set. I'm trying to use the Weiertrass $M$ - test to show that the ...
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### the limit of the infimum of a sequence of bounded functions that converge uniformly is equal to the infimum of the limiting function

prove $\lim_{n\to\infty}inf [f_n(x)|x\in E]=inf[f(x)|x\in E]$ where $f_n$ are bounded functions of a set $E\subset R$ that converge uniformly to a function $f$. I've looked at similar proofs on the ...
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We wish to prove that the sequence of functions $f_n(x)=\frac{1+\cos^2(nx)}{\sqrt{n}}$ converges uniformly to $0$ on $\mathbb{R}$. Scratch work: $|f_n(x)|=|\frac{1+\cos^2(nx)}{\sqrt{n}}|\le|\frac{2}{\... 0 votes 2 answers 29 views ### (Locally) Uniform convergence of the given series Determine the locally uniform convergence of the series$\sum_{n=1}^{\infty} f_n(z)$on$V=\mathbb{C}\setminus\{-n : n\in\mathbb{N}\}$, where $$f_{n}(z) = (-1)^n \frac{1}{z+n}.$$ Is$f_n$also uniform ... • 73 0 votes 1 answer 67 views ### Is uniform convergence required for a continuous limit function? Consider the sequence$(f_{n})_{n=1}^{\infty}$of continuous functions on$I = [0, \infty)$defined recursively by$f_{1}(x)=x, f_{n}(x)=x+\int_{0}^{x}f_{n-1}(t)\sin(x-t) dt, \forall n\geq 2$. This ... • 35 0 votes 0 answers 82 views ### Convergence in$L^2$doesn't imply uniform convergence I am working on implications between different types of convergences in series of functions. I think that$L^2$convergence doesn't imply uniform convergence, however I cannot find a counter-example. ... • 1,263 0 votes 1 answer 53 views ### Issues with theorem 7.15 from Rudin's PMA. Some prerequisite definitions and theorems. Given a metric space$X$, let $$\mathscr{C}(X) = \{f: X \rightarrow \mathbb{C} \; | \; f \text{ bounded and continuous.}\}.$$ Next we have the familiar ... 2 votes 1 answer 93 views ### Proving$f_n(x) = x^n$for$x \in [0,1]\$ is not a uniformly Cauchy sequence

I know this can be argued more succinctly by citing theorems on uniform convergence. I have also seen answers to this question on this page, but I made a somewhat different (I think) attempt of my own....
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