Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

1
vote
3answers
88 views
+50

Finding Borel sets

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $$f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$$ Is $f$ Borel-measurable? I followed the answer given here ...
8
votes
1answer
115 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
4
votes
3answers
127 views

If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$

Let $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0,$ where $f$ is any integrable function. If simeone is allowed to use the Mean Value thorem for integrals, the proof is at hand. ...
0
votes
0answers
14 views

Algebraic closure of p-adic rationals, $\overline{\mathbb Q}_p$, and its completion, $\Omega_p$, are not locally compact

Trying to show $\overline{\mathbb Q}_p$ and $\Omega_p$ are not locally compact. I can prove it by showing that the unit sphere is not locally compact. That is to say, any sequence on the unit sphere ...
0
votes
0answers
7 views

Total Variation of Subintervals

Studying functions of bounded variation, the following exercise showed up: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is ...
2
votes
0answers
22 views
+50

Existence of the $\Omega$ set in “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”

In the proof of Lemma 4.3 in [1], they claim the following: Let $U$ be a subspace of $\mathbb{R}^{m\times n}$ with dim$(U)=d$ and let $\delta>0$. Then, there exists a set $\Omega\subset\mathbb{R}^{...
1
vote
1answer
25 views

number of zeroes of arbitrary function

Sorry if I misused/mixed up some maths terms. I barely know any maths lingo, especially not in English. I was thinking about programmatically solving equations (or rather, approximating their roots), ...
1
vote
2answers
26 views

For some $c \geq 0$ $\text{sup} \ \{ c \cdot f(x): \text{some domain of $x$} \}$ = $c \cdot \text{sup} \ \{f(x): \text{same domain of $x$} \}$ [duplicate]

How would you formally justify this? Or is it just notationally obvious? (As opposed to 'conceptually' obvious, which is never an excuse in mathematics.) Edit: For some $c \geq 0$ $\text{sup} \ \{ c ...
1
vote
3answers
18 views

Partial derivative of the real part of a function

I'm trying to understand the mathematical reasoning behind the example provided in this question. If we have $$z = Ae^{i(\omega _{o}t+\phi )}$$ and define $$x = Re (z),$$ then why is it that $$\...
0
votes
0answers
10 views

Explanation of Nested Interval Theorem

I have a question relating to the theorem about nested intervals. I understand it until the last expression where it is stated that the interval $[a,b]$ is included in the intersection of those ...
1
vote
1answer
41 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
0
votes
0answers
60 views

Arc-length parametrisation

I can not understand how the concept of arc length works. We define $a(t)=\int_{0}^{t}\mid\gamma^{'}(x)\mid dx$ for some curve $\gamma: I \rightarrow\mathbb{R}^2$. If we then normalise the curve we ...
4
votes
1answer
896 views

nondecreasing rearrangement is equimeasurable

Two functions $f(x)$ and $g(x)$ are called equi-measurable if $m(\{x:f(x)>t\})=m(\{x:g(x)>t\})$. Nondecreasing rearrangement of a function $f(x)$ is defined as $$f^*(\tau)=\inf\{t>0:m(\{x:f(x)...
-1
votes
1answer
26 views

solution of system

Does the solution of the system satisfy $z=v=0$ and why? \begin{align} -a^{2}z-z_{xx}+i\beta a v&=0 \\ -a^{2}v-v_{xx}-i\beta a z&=0 \end{align} where the system is defined on $(l,L)$, with $...
0
votes
1answer
33 views

$f$ is Lipschitz and $X$ have measure zero. Show that $f(X)$ has measure zero too.

If $f[a,b]\to\mathbb{R}$ is Lipchitz and $X\subset [a,b]$ has measure zero show that $f(X)$ has measure zero too. What I did: As $X$ has measure zero, $\forall \epsilon>0$ there exists a ...
-6
votes
0answers
44 views

‎If ‎$‎\lim‎ f(x) = L‎$ as ‎$‎x‎\rightarrow ‎+‎\infty‎$‎, then $‎\lim‎ f(c x) = L‎$ as $‎x‎\rightarrow ‎+‎\infty‎$. [on hold]

‎Let ‎‎$‎f‎$ ‎be a‎ ‎real-variable ‎function ‎such ‎that ‎‎$‎‎‎‎‎\lim‎‎ f(x) = L‎$ as ‎$‎x‎‎\rightarrow ‎+‎\infty‎$ where $‎L\in‎\mathbb{R}‎$‎. ‎Also, let‎ $‎c‎>0‎$ be a constant. My question is:‎ ‎...
1
vote
1answer
25 views

A Lipschitz function is $C^1$?

I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
-1
votes
1answer
23 views

What's the difference between the operator norm and the sup norm

What's the difference between the operator norm and the sup norm over $C[0,1]$. a.k.a $\left\lVert x\right\rVert_\infty$ vs $\left\lVert x\right\rVert_{op}$
0
votes
3answers
74 views

Studying the character of $\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$

I have to study the character of this series $$\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$$ with $\alpha$ a real parameter. Considering the Cauchy condensation test, the equivalent ...
4
votes
2answers
342 views

Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.

Let $A\subset \mathbb{R}$ be of positive Lebesgue measure, i.e. $\mu(A)>0$. Is it then true that $\mu(\mathbb{R}\setminus (A+\mathbb{Q})) = 0$? I am quite sure that if $\mu(A)>0$, then $A-A$ ...
3
votes
1answer
36 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
-2
votes
0answers
15 views

Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
1
vote
0answers
37 views

$(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers. [duplicate]

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that $$ (x+y)^r \le x^r+y^r? $$ In all the examples I considered, ...
-2
votes
1answer
30 views

Conditionally convergent series, true or false

It is given that the series $\sum_{n=1}^{\infty}a_n$ is convergent but not absolutely convergent and $\sum_{n=1}^{\infty}a_n=0$ denote by $S_k$, the partial sum $\sum_{n=0}^{k}a_n,~k=1,2,...$, then $...
3
votes
2answers
55 views

Show power series converges for every $x$.

Let $$f(x) = 1 + a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...$$ be a solution of the differential equation $f'(x)=xf(x).$ Now I need to explain that the power series that define $f(x)$ converges ...
-3
votes
0answers
20 views

Exercise about ordinary differential equation

I have some problem about ordinary differential equation. Please help me solve and explain it. Thank you very much. Good health! Problem 1. Suppose $y(x)$ is a solution of the equation $$y''-2y'+y=2e^...
0
votes
0answers
12 views

mean of a field along the line

What is the proof of a mean field along any line? Or how can we define it? i.e \begin{equation} a=\int_0^1 b \, dx \end{equation} where $a$ is the mean of $b$ and $b(0)=b(1)=0$.
0
votes
1answer
55 views

$I_{n}=\left[a+\frac{(k_n-1)}{2^n};\ a+\frac{k_{n}}{2^n}\right]$

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : $$\...
0
votes
2answers
93 views

Show that there exists $m\in\mathbb{N}$ such that $a+\frac{m}{2^n}\geq b$

Let $ \mathcal{P} \subset \mathbb{R}$, $\ \mathcal{P}\neq \emptyset $ et let $b$ an upper bound of $\mathcal{P}$ Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : $$\exists\ m\...
3
votes
2answers
73 views

Prove convergence of $ \sum_{n=1}^{\infty} a^{\ln n}$ for $0<a<\frac{1}{e}$ [duplicate]

Prove the convergence of the series: $$\sum_{n=1}^{\infty} a^{\ln n},\,\text{for} \,\,0<a<\frac{1}{e}.$$ Attempt. I have proved the non-convergence in the case $a\geq 1/e$ (using the ...
0
votes
1answer
42 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
0
votes
0answers
31 views

Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...
0
votes
1answer
31 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
-1
votes
0answers
49 views

$\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$

Prove that exist $c>0$ such that $\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$ for all $f$ $\in$ $C^2(0,1)$. Maybe that'll help, we can use similar statement about supremums: $\sup_{(0,1)}|...
1
vote
0answers
26 views

Can we write the following limit -equality?

We know: 1-) The following condition is valid ONLY with zeros (any) for $\,\,\,\,\,$ $0<x<2$ and $x\in \mathbb R $: $$f(x_o)=g(x_o)=0 \,\,\,\,\ ⇔ \,\,\,\,\ f(2-x_o)=g(2-x_o)=0$$ 2-) ...
1
vote
0answers
44 views

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$. Since the question is asking for a sequence of continuous ...
3
votes
3answers
33 views

$g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$, which of the following option is true?

Suppose $f$ is an increasing real valued function on $[0,\infty)$ with $f(x)>0$ for all $x$ and let $g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$; $0 < x <\infty$, Then which of the following are true:...
0
votes
1answer
48 views

series that diverges

We consider the sequence $$(u_n)_{n\in\mathbb{N}}$$ defined by $0<u_0<1$ and $u_{n+1}=u_n-u_{n}^2$ for all $n\in\mathbb{N}.$ I want to prove that the serie with general term $\ln(\frac{u_{n+1}...
1
vote
1answer
30 views

$[F]_p\le [f]_1[g]_p$ for $1\le p\le\infty$

For real-valued functions $f$ and $g$ on $(0,\infty)$, let $$F(x)=\int_0^\infty f\left(\frac{x}{y}\right)g(y)\frac{dy}{y}$$ If $1\le p\le\infty$, set $$[h]_p=\left(\int_0^\infty |h(x)|^p\frac{dx}{...
0
votes
1answer
29 views

$\liminf_\limits{n\to\infty}1_{A_n}(x)=1$ $\implies$ $\lim_\limits{n\to\infty}1_{A_n}(x)=1$?

Source: Partial proof from textbook: I've omitted the case where $x\in A^c$ as it's not relevant. I've also highlighted the part I'm having trouble with in blue. Here is my attempt at explaining ...
0
votes
0answers
23 views

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? [on hold]

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? Are they just the same thing?
0
votes
0answers
24 views
-2
votes
0answers
47 views

$\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$ [on hold]

Let $f \colon [0,\infty)\to\mathbb{R}$ be a strictly decreasing, continuous function. Suppose $\lim_{x\to\infty} f(x) =0$. Prove that $\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$.
0
votes
1answer
35 views

How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
0
votes
1answer
24 views

Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \geq N$} is compact.

Define the set $A \subseteq {\ell}^2$ by $A = ${{$x_n$} $\in {\ell}^2 : \sum_{n=0}^{\infty}(1+n){|x_n|}^2 \leq 1$} i) Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \...
0
votes
1answer
62 views

Is this product strictly positive?

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(p-1)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
0
votes
0answers
35 views

Is a partial differential equation satisfied after reduction to a subspace?

I have a $n$th-order non-linear partial differential in $m$-real variables $x_1,x_2, \ldots, x_m$. Assume a function $f$ satisfies this differential equation. I denote this by $$D f(x_1, x_2, \ldots, ...
0
votes
0answers
11 views

$\overline{lim}$ the set of cluster points

I am currently reading an article where the author has the following statement. " $\overline{\lim}_{t\to\infty} u_t$ is the set of cluster points of the sequence $u_k \subset U$ which is nonempty ...
0
votes
1answer
17 views

Is induction the correct approach here?

Let $a_1, a_2, \dots, a_n$ (real numbers) be such that $$a_1 - a_2/3 + \dots + (-1)^{n - 1}a_n/(2n - 1) = 0.$$ Prove that $$f(x):= a_1\cos(x) + a_2\cos(3x) + \dots + a_n\cos([2n - 1]x) = 0,$$ for ...
0
votes
1answer
22 views

Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...