In every Arzela Ascoli proof you see the following: Let $S = \mathbb{Q} \cap [a,b]$, where $[a,b]$ is an interval in $\mathbb{R}$, then $S$ is a countable dense subset and there exists a finite number of points $\{x_1, \ldots, x_k\}$ such that $\forall x \in [a,b], |x - x_j| < \delta$, ...

I am trying to work through a proof on a modification of the ascoli theorem that is supposed to hold in ZF (even without assuming countable choice). My problem is within the part (1) $\Rightarrow$ (2b): Let $F$ be a set of continous functions from $\mathbb{R} \to \mathbb{R}$ and let each sequen...

Let $D= \{f_k(x) = \sin kx/(1+k);\quad k= 0,1,2,\dots\quad x \in [0,1]\}$. Show that $D$ is compact. I know that I should approach this kind of question by Ascoli-Arzela theorem (proving $D$ is equicontinuous, bounded and closed). For equicontinuity, do we always use $3\epsilon$ technique? Thus...

When applying the Arzela Ascoli theorem, I am little bit confused as to what needs to be done to show that a set of functions in $C_0([a,b])$ is closed. Sifting through MSE I found a claim that : "a set of functions is closed if it contains the pointwise limit and converges uniformly to it...

I've been looking for a proof of one particular direction of this theorem for metric spaces. I've looked online, but everyone seems to use different terminology/notation to state the theorem, so I'd like to ask for an outline of a proof specific to my text's version, which my text "leaves to the ...

I am working on the following exercise from Royden's Real Analysis (Chapter 10, Section 10.1 on the Arzela-Ascoli Theoerem): Let $S$ be a countable set, and $\{ f_{n} \}$ a sequence of real-valued functions on $S$ that is pointwise bounded on $S$. Show that there is a subsequence of $\{ f_{n}...

For a closed, bounded interval $[a,b]$, let $\{ f_{n}\}$ be a sequence in $C[a,b]$. If $\{f_{n}\}$ is equicontinuous, does $\{f_{n}\}$ necessarily have a uniformly convergent subsequence? I would think not, because according to the Arzela-Ascoli Theorem, $\{f_{n} \}$ also needs to be uniformly ...

I am attempting to solve the following problem: Let the sequence of continuous functions $\{x_{n}(t) \}_{n=1}^{\infty}$, $0 \leq t < \infty$ be uniformly bounded on $t \in [0, \infty)$ and on each bounded interval $[0,A]$ also satisfy the Lipschitz condition: $\forall t_{1}, t_{2} \in [0,A]$...

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in $L^q(X)$ for every bounded $X\subseteq\mathbb{R^d}$ and every $q$ such that $2 \leq q<\frac{2d}{d-2s}$...

Let $F$ be a set of continuous functions on $[0,1].$ Assume that every sequence in $F$ has a subsequence that converges uniformly on $[0,1].$ Prove that there is a uniform bound for all functions in $F,$ and that $F$ is equicontinuous. This is the converse of the Arzela-Ascoli theorem. I belie...

I am looking for a reference in the literature of the following corollary of Ascoli-Arzelà's theorem : $K\subset\mathbb{R}^n$ is compact. The set $\{f:\mathbb{R_+}\to K \ | \ f \text{ is }1- \text{ Lipschitz continuous }\}$ is compact for the compact convergence topology. Thanks!

I'm stuck on the last step of a real analysis/advanced calculus problem and could really use some help. The problem is as follows: Let $f_n$ be continuously differentiable on $[0,1]$ satisfying, for all $n \geq 1$ , $$\hspace{.5cm} |f_n'(x)| \leq \frac{1}{\sqrt{x}}, \hspace{.3cm} 0 < x \leq 1,...