0
votes
1answer
40 views

Let m ∈ N. Define the relation ≡^ on Z by a ≡^ b for a, b ∈ Z if and only if a ≡ ±b (mod m).

(In other words, the relation ≡^ holds if either a ≡ b (mod m) or a ≡ −b (mod m).) Prove that the relation ≡^ on Z is transitive. ======= I believe there are 3 properties that it must meet ...
2
votes
1answer
34 views

Is my proof ok? Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive.

Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive. If $A$ is congruent to $B$ mod $m$ then $A - B = k m~~$ (1) If $B$ is congruent to $C$ mod ...
2
votes
0answers
79 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
6
votes
1answer
50 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
2
votes
1answer
50 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
1
vote
1answer
39 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
6
votes
1answer
106 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
1
vote
2answers
24 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
1
vote
1answer
23 views

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
1
vote
0answers
39 views

Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
0
votes
1answer
38 views

Non-increasing Monotone Sequence Convergence Proof

My goal is to prove the monotone convergence of a non-increasing sequence of real numbers. There are some steps in the proof that I'm not sure about. The question: If $S$ is a non-increasing sequence ...
0
votes
1answer
39 views

If a sequence of functions converges uniformly, then its limit is bounded

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of functions defined on $[a,b]$ Assume that each $f_n$ is bounded, so $|f_n| \le M_n$ for all $x \in [a,b]$. If $\{f_n\}_{n=1}^{\infty}$ converges uniformly ...
3
votes
0answers
51 views

Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$. Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges ...
4
votes
1answer
57 views

Proving for all polynomials $p(x)=p_0+p_1x+…+p_dx^d$, $ \ \sum^\infty_{n=1}p(a_n)$ converges iff $p_0=0$

The series $\displaystyle\sum^\infty_{n=1}a_n$ converges absolutely. Prove that for every polynomial $p(x)=p_0+p_1x+...+p_dx^d$, $\ \displaystyle\sum^\infty_{n=1}p(a_n)$ converges iff ...
0
votes
1answer
32 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
37 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
33 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
48 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
60 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
109 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
70 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
36 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
79 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
78 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
1answer
76 views

Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$

I'm reading Real Analysis by Royden 4th Edition. The entire problem statement is: Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ ...
0
votes
3answers
120 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
203 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
62 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
89 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
129 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
1answer
53 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
2answers
119 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
188 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 ...
4
votes
1answer
393 views

Uniform Convergence of Series involving sin x

True or False: The series $$\sum_{n=1}^\infty \frac{\sin x}{1+n^2x^2}$$ converges uniformly on $[-\pi,\pi]$. Appreciate comments on attempt below, especially if it's incorrect! Attempt at ...