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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.

4
votes
6answers
55 views

Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
1
vote
1answer
13 views

Uniform convergence about the function $h$

Consider the sequence of functions $$h_n(x)=\begin{cases}2nx&\text{if}\;x \in [0,\frac{1}{2n}]\\\\2-2nx&\text{if}\;x \in [\frac{1}{2n},\frac{1}{n}]\\\\0&\text{if}\;x\geq \frac{1}{n} \end{...
4
votes
2answers
355 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 $A-A$ ...
0
votes
0answers
43 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 \...
1
vote
2answers
888 views

Describe the sigma algebra generated by singleton subsets

Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases: i) $X$ is countable ii) $X$ is ...
1
vote
1answer
50 views

Show $e^{a-b}=\frac{e^a}{e^b}$ for every $a,b \in \mathbb{C}$.

I would appreciate to have my proof verified Show $e^{a-b}=\frac{e^a}{e^b}$ for every $a,b \in \mathbb{C}$. As is already known that $e^{a+b}=e^{a}e^{b}$ for every $a,b \in \mathbb{C}$, then for ...
0
votes
1answer
19 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 ...
4
votes
2answers
46 views

Infinite closed set in $\mathbb{R}$ can be generated by its countable subsets

Suppose $F\subset\mathbb{R}$ is a infinite closed set. Prove that there exists a countable subset $E\subset F$ such that $\bar{E}=F$.
6
votes
4answers
5k views

Simplifying square of integral in general

For a real-valued function $f=f(x)$, over the real variable $x$, with the following integral $$ \left[ \int_{a}^{b} f(x)dx \right]^{2}, $$ is there a known general method/approach to handle this as ...
6
votes
4answers
195 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
4answers
38 views

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$ Firstly I wanted to calculate $\int \sin (t^2) dt$ and then use $x$ and $\sqrt {x^2+1}$. But this antiderivative not exist so how can I do ...
-1
votes
0answers
45 views

Help calculate the limits

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
1
vote
1answer
46 views

Application of Jensen's Inequality to non-negative integrable function

I am reading a book, where it uses the following result. Can someone help me to derive the result? I know i have to use Jensen's inequality here, but not sure how to get the final result. Here is the ...
-1
votes
1answer
29 views

Every isometry is Lipchitz-continuous

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, $f:(X,d_X) \rightarrow (Y,d_Y)$ be an isometry. Then $f$ is Lipchitz-continuous. Attempt: Suppose that $f$ is an isometry. Then for all $x_1,x_2$ in $...
-1
votes
1answer
18 views

A uniformly continuous function with the supremum metric…

Question: Suppose that $C([0, 1])$ is the metric space of all continuous real-valued functions on $[0, 1]$, with the metric $d(f, g) := \sup_{x \in [0, 1]}|f(x) - g(x)|$. Let $f \in C([0, 1])$ such ...
0
votes
3answers
31 views

Why is it possible to calculate multivariable limits using polar coordinates?

Why is it possible to calculate multivariable limits using polar coordinates? Let's say I'm looking for some $\lim_{(x,y) \to (0,0)}$ and I'm substituting $x = r cos\theta$ and $y = rsin\theta$ so ...
0
votes
0answers
29 views

Is the inf-convolution of two continuous functions continuous?

For two continuous functions $f$ and $g$ defined on a normed space $E$ taking values in $[-\infty,\infty)$, let $f\square g: x\mapsto \inf\{f(y)+g(x-y): y \in E\}$. Here I do not assume any ...
4
votes
1answer
36 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}^{...
2
votes
1answer
47 views

Multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$

How can I calculate the multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$? I'm new to this and I've seen a couple of examples, where it is possible to limit the fraction by a ...
0
votes
1answer
42 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!
1
vote
2answers
60 views

Calculate $\int e^{2x}(\cos x)^3 dx$

Calculate $$\int e^{2x}(\cos x)^3 dx$$ My try: Firsty I tried to use integration by parts but then I got: $$\int e^{2x}\cos^3(x) dx=...=\frac{1}{2}\cos^3(x) e^{2x}+\frac{3}{2}\left(\int e^{2x} \sin(...
-1
votes
0answers
65 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)}|...
0
votes
1answer
24 views

Directional derivative equivalent definition

Let $f=f(x_1,\dots,x_n)$ be a scalar function defined on some open subset $U\subset\mathbb{R}^n$. Given an unit vector $v=(v_1,\dots,v_n)$ and a point $x_0\in U$, the directional derivative of $f$ at $...
0
votes
0answers
18 views

Inverse image of a variable under composite mapping

Let $f: X \to Y$ and $g: Y \to Z$. ($.^{-1}$ denotes inverse image). Is the set $f^{-1}(y)$ equal to the set $(g\circ f)^{-1}(g(y))$ for all $y$ ?
1
vote
0answers
36 views

Show that $f(x) = {1\over x^n}$ is continuous in its domain, $n\in\Bbb N$

Let $n\in\Bbb N$. Show that $$ f(x) = {1\over x^n} $$ is continuous in its domain. I've recently shown that $g(x) = x^n$ is continuous everywhere in $\Bbb R$. Now I want to do the same for $1/x^n$,...
1
vote
1answer
13 views

Show there are basis vectors such that $Dg(a)(e_{j_k})$ spans $\mathbb{R}^p$

Given $p<n$, $g:\mathbb{R}^n\to\mathbb{R}^p$ a $C^1$ function with $g(a)=0$ and $Dg(a): \mathbb{R}^n\to\mathbb{R}^p$ surjective. And using the identification $\mathbb{R}^p\cong \mathbb{R}^p\times \{...
-6
votes
0answers
46 views

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

‎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:‎ ‎...
2
votes
1answer
922 views

Convergence in measure of product of convergent sequences

Let $(X,\Sigma,\mu)$ be a finite measurable space ($\mu(X)<\infty$). Suppose $f_n \xrightarrow{\mu} f$ and $g_n \xrightarrow{\mu} f$, prove that $f_ng_n \xrightarrow{\mu} fg$ I'll write what I ...
0
votes
1answer
19 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 ...
1
vote
3answers
95 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
126 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(\...
1
vote
1answer
31 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
27 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 ...
2
votes
3answers
36 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
17 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 ...
2
votes
1answer
44 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
897 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
31 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
38 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 ...
1
vote
1answer
29 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
26 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
75 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 ...
3
votes
1answer
40 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
17 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
38 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
31 views

Conditionally convergent series, true or false [on hold]

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
57 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 ...
-4
votes
0answers
23 views

Exercise about ordinary differential equation [on hold]

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
14 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 $b(0)=b(1)=0$.