0
votes
1answer
28 views

The proportion of $\omega$s in $A$ converges almost surely to $P(A)$

Let $A$ be an event in $(\Omega,\mathcal{F},P)$. We generate independent inquiries from $\Omega$ in accordance to $P$. Show that the proportion of $\omega$s in $A$ converges almost surely to ...
2
votes
1answer
26 views

Totally continuous implies bounded

Consider a separable, reflexive Banach space $V$. We define the mapping $A: V \rightarrow V^{*}$ as totally continuous if it is continuous as a mapping $(V, \text{weak}) \rightarrow (V^{*}, norm)$. I ...
1
vote
1answer
31 views

Domination $\Rightarrow$ $0$ equality

Let $\phi \in C^{\infty}_c(\mathbb{R})$. In class, my teacher said that the dominated convergence theorem (DOM) may be used to prove that $$ \lim_{\epsilon \to 0^+} \int_{-\epsilon}^{\epsilon} \! \log ...
1
vote
1answer
32 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
0
votes
1answer
58 views

Proof that a sequence has a convergent subsequence

I have a bounded sequence $a(n)$. We consider the set of all the values of $a(n)$ and let $M$ be the supremum of this set (without being one of its elements). Now we want to show that there is a ...
2
votes
1answer
78 views

Theorem 9.5 Cauchy Condition for uniform convergence of series - Math Analysis 2nd ed - Apostol

Theorem 9.5 (Cauchy condition for uniform convergence of series) The infinite series $\sum f_n(x)$ converges uniformly on $S$ if, and only if, for every $\epsilon>0$ there is an $N$ such that ...
2
votes
1answer
68 views

Show that $f_n1_{A_n}$ convergences in mean

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions on that measurable space and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)$ converge in ...
1
vote
1answer
31 views

On (absolute) convergence of $f_c:= c + \sum_{n=0}^{+ \infty} \frac{a_n}{n+1}x^{n+1}$

Let $R> 0$ and let $g: (-R,R) \longrightarrow \mathbb{R}$ be given by the convergent power series $$g(x):= \sum_{n=0}^{+ \infty} a_nx^n$$ for $|x| < R$. Let $c \in \mathbb{R}$ and let $f_c: ...
0
votes
0answers
65 views

Parameter integral and continuity (Theorem of Lebesgue)

I already kept myself busy with a proof concerning the Theorem of Lebesgue and differentiation of a parameter integral. Unfortunately I did not get an answer there yet. Now my task is nearly the ...
0
votes
0answers
65 views

Theorem of Lebesgue and differentiation of a parameter integral

Let $(a,b)\subset\mathbb{R}$ be an interval and let $\left\{f_t\colon\Omega\to\mathbb{R}\right\}_{t\in (a,b)}$ be a family of measurable functions on the measurable space ...
0
votes
3answers
76 views

Help with Pointwise and Uniform Convergence in Metric Spaces

I am having a bit of difficulty understanding uniform convergence and would also like to check my understanding of pointwise convergence. Using the example of $f_n$(x) = $x^n$ on (-1,1), I found the ...
2
votes
2answers
192 views

Verification of proof of the Sequence of Arithmetic Theorem

Suppose $\left\{b_{n}\right\}$ is a sequence of real numbers which converges to $M$, so that $b_{n} \neq 0$ for each $n$, and $M \neq 0$. Prove that the sequence $\{ \frac{1}{b_n} \}$ converges to ...
4
votes
0answers
39 views

General and basic question about convergence of a series

Let $(a_{i,j})_{i,j=1}^n$ be a sequence of real numbers such that the following series converges $$ S = \lim_{n\to\infty}\sum_{i=1}^n\sum_{j=1}^na_{i,j} $$ It is known that for each $i$th the ...
4
votes
1answer
59 views

For this periodic continuous $g:\Bbb R\to \Bbb R$, and $f_n(x):=g(x/n)$, does $\{f_n\}_{n=1}^\infty$ converge uniformly?

I can not find a counterexample although I have the feeling it is not true. Let $\ g: \mathbb{ R} \rightarrow \mathbb{R}$ continuous function $ \forall x \in \mathbb{R} \ g(x+1) = g(x)$ $g(0) = 0$ ...
0
votes
2answers
88 views

Prove $\left(\frac{1}{n}+\frac{(-1)^n}{n^2}\right)$ converges to $0$ as $n\to\infty$

Using the formal definition of convergence of a sequence, show that the sequence converges to 0 as n tends to infinity. So we want to show that for every $\epsilon>0$, there exists $N$ such that ...
2
votes
1answer
127 views

(Edited Duplicate) Let <$x_{n_n}$> be a sequence of positive real number that has no convergent subsequence. Show lim $x_n$ = +$\infty$

Proof: Suppose $(x_{n})_n$ is a sequence of positive real numbers which has no convergent subsequence. By contradiction we have $(x_{n})_n$ is not bounded, for if it was then it would admit a ...
2
votes
0answers
43 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
0
votes
1answer
63 views

Showing a series is not uniformly convergence

Suppose you want to show a series does not converge uniformly on some interval. If you know the point wise limit is $f$, and you can show the $\sup |f_{n} - f|$ does not go to zero on your interval, ...
1
vote
1answer
150 views

Alternative proof or verification of given proof of convergence in probability

I am asked to show that if $X_n \rightarrow c$ in probability and if $g$ is a continuous function, then $g(X_n) \rightarrow g(c)$ in probability for a statistics homework problem in a section titled ...