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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.

3
votes
1answer
26 views

Calculating $\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + …+\frac{\sin(2\sqrt n)}{n\sqrt n\cos\sqrt n}\right)$

Using the trigonometric identity of $\sin 2\alpha = 2\sin \alpha \cos \alpha$, I rewrote the expression to: $$\displaystyle{\lim_{n\to\infty}}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + ...+\...
0
votes
0answers
14 views

How can we prove that this integral converges?

If we have integral in the form $\int_0^\infty \frac{1}{(r+a \exp (-xt) )]\sqrt{(1+(a \exp (-xt))^2)}} \ dt $ and if we take difference of such two integrals with the same $r$ and different $a$ and $x$...
2
votes
4answers
67 views

Calculating $\displaystyle {\lim _{n\to\infty}}\frac{1+2+3+…+n-1}{n^2}$

My first attempt was using limit arithmetic, but it fails because one of the operands is infinite, so that didn't work. I then tried using the squeeze theorem: $$b_n = \frac{1+2+3+...+n-1}{n^2}$$ $$...
1
vote
1answer
22 views

Equivalence with measurable functions and spaces

i have been reading Follands Real Analysis Book and i got stuck with one exercise. It says that if $\lambda(X)$ is finite and $(X,M,\lambda)$ is a measure space and $(X, \overline{M}, \overline{\...
0
votes
1answer
13 views

Proving that the equivalence of paths is an equivalence relation.

The definition of equivalent paths is as follows : Two paths $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ are equivalent if there exist a $C^{1}$ bijection $\phi: [...
0
votes
2answers
32 views

Showing that 2 paths are not equivalent.

The definition of equivalent paths is as follows : Two paths $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ are equivalent if there exist a $C^{1}$ bijection $\phi: [...
0
votes
1answer
9 views

Proof Check: Outer Regularity of Lebesgue Measure

I am trying to prove that for any bounded set $A$ in the borel $\sigma$ algebra, that for the Lebesgue measure $m$ $$ m(A)=\inf\{m(U)|U\text{ is open and }A\subseteq U\} $$ here is my attempt. Let $...
0
votes
1answer
32 views

Vanishing of the integration along vertical line

2- Show that if $C$ is a vertical line segment $c \leq y \leq d,$ and if $F$ is a function of 2 variables defined on $C$, then $$\int_{C} F(x,y)dx = 0$$. I understand that the integration is ...
1
vote
0answers
16 views

For $g(x) = \int_0^\infty \frac{1}{x+y} f(y) \, dy$, Show $m\{ x \in (0,\infty) : g(x) > \lambda \} \le 1/\lambda \cdot \lVert f \rVert_{L^1}$

Q: For $x \in (0, \infty)$ let: \begin{align*} g(x) &= \int_0^\infty \frac{1}{x+y} f(y) \, dy \\ \end{align*} Show that for $f \in L^1(0,\infty)$: \begin{align*} m\{ x \in (0,\infty) : g(...
0
votes
0answers
18 views

Question regarding the use of the Implicit function theorem

Let $f: U \subseteq \mathbb{R}^m \rightarrow \mathbb{R}^{m+1}$ be a $C^k$ function. Suppose $$ f(x)=(f_1(x), f_2(x), \cdots, f_{m+1}(x)) \quad \text{where} \quad det \; \left ( \frac{\partial ...
0
votes
0answers
9 views

Multidimensional Correlated Geometric Brownian Motion, finding exact form of the matrices

My goal is to understand the dimensions of the matrices involved, so I am initially writing things as column vectors, and defining all the dimensions. I am working with the following setup: ...
1
vote
2answers
51 views

Rudin Principles of Mathematical Analysis Chapter 10, Exercise 8

I'm working on exercises of chapter 10 in Baby Rudin. I refer to R. Cooke's solutions manual to Baby Rudin while I'm solving those exercises.(https://minds.wisconsin.edu/handle/1793/67009) But I ...
1
vote
1answer
18 views

The integration of the norm of the derivative of 2 equivalent paths are equal.

Let $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ be 2 equivalent paths. Prove that $$\int_{a}^{b} ||f '(t)|| dt = \int_{c}^{d} ||g '(t)||dt. $$ The definition of ...
0
votes
1answer
20 views

Proof that l2 has a countable and dense subset

this is my first question and I hope I don't make any relevant mistakes. For a little bit of context, in my real analysis homework I have the following problem. Show that the subset D $\subset$ $l_2$...
2
votes
0answers
20 views

For $f \in L^p(0,\infty)$, $g(x) = \int_0^\infty \frac{1}{x+y} f(y) \, dy$, show $\lVert g \rVert_{L^p} \le A_p \lVert f \rVert_{L^p}$.

For $f \in L^p(0,\infty), 1 \le p \le \infty$ \begin{align*} g(x) &= \int_0^\infty \frac{1}{x+y} f(y) \, dy & x &\in (0, \infty) \\ \end{align*} Show that $\lVert g \rVert_{L^p} \...
3
votes
1answer
59 views

Solve the integral $\int_{0}^{1} \frac{x^\frac{1}{10}}{1+e^x}dx$

While solving a certain problem, I got stuck at this integral $$\displaystyle \int_{0}^{1} \frac{x^\frac{1}{10}}{1+e^x}dx$$ I tried elementary methods but nothing works. The last option would be a ...
0
votes
1answer
61 views

Does the Integral $\int_{-\infty}^{\infty} e^{-iwx}\cos(kx) \ dx$ converge?

I am trying to determine whether or not the integral $$\int_{-\infty}^{\infty} e^{-iwx}\cos(kx) \ dx$$ converges, to determine if $\cos(kx)$ has a Fourier transform. Intuition tells me it does not ...
0
votes
0answers
40 views

Real Analysis: Derivative of $sin(x_1)+cos(x_2)$ at (0,0)

How do you prove that $sin(x_1)+cos(x_2)$ is differntiable at 0 using the following definition of differentiability? $f:U \rightarrow R^m$ is differentiable at $x$ if there exists a linear map $T:...
0
votes
1answer
46 views

Proving a Line-integration along a parametrized curve identitiy.

(this question were asked after studying line integrals) 1- Show that if $C$ is the graph of $y=f(x)$, $a \leq x \leq b,$ and if $F$ is a function of 2 variables defined on C, then $$\int_{C} F(x,y)...
0
votes
1answer
25 views

$\epsilon-\delta$ proof that $\lim_{(x,y)\to(c,0)} \frac{\sin(x^2y)}{x^2-y^2} = 0$ when $c \neq 0$

I'm trying to prove by definition that, for $c \neq 0$, $$\lim_{(x,y)\to(c,0)} \frac{\sin(x^2y)}{x^2-y^2} = 0$$ I started this way: $$\bigg|\frac{\sin(x^2y)}{x^2-y^2}\bigg|= \frac{|\sin(x^2y)|}{|x^...
1
vote
2answers
51 views

Proving $\lim _{n\to\infty}\frac{1}{ a_n} \neq \alpha$ given $\lim _{n\to\infty}a_n = 0$ and…

Given: $$\displaystyle {\lim _{n\to\infty}}a_n = 0\\\alpha \in \mathbb{R}\\a_n \neq 0$$ I'm trying to show: $$\exists \mathcal{E} > 0| \exists N \in \mathbb{N}|\forall n > N:$$ $$\left| \frac{1}{...
0
votes
1answer
13 views

proof verification: Proving an upper bound for a uniformly convergent sequence of functions on $\mathbb{R}$

Assume $f_n: \mathbb{R}\to\mathbb{R}$ is a sequence of functions that converges uniformly to $f$. Assume that there exists $M>0$ such that for all $n\in \mathbb{N}$ and $x\in \mathbb{R}$ one has $|...
0
votes
1answer
46 views

What kind of numbers are inside a generating open interval of the Borel $\sigma$-algebra? [on hold]

If it is enough to have all open intervals (a,b) with end points $a$ and $b$ belonging to the rational numbers, a < b, in order to generate a Borel $\sigma$-algebra on $\mathbb{R}$. Asked here: ...
1
vote
1answer
9 views

Derivate of inverse of composite function

I'm very confused, and this is probably a stupid question. I want to calculate $ \frac{d}{dx} f^{-1}(g^{-1}(x))$. However, I get two seemingly different results taking two different approaches. I. $\...
0
votes
0answers
25 views

Proof set of content zero has void interior

Given a set of content zero $A \subset R^{n}$, how can be proved that the interior of $A$ is void?
0
votes
6answers
52 views

Prove that a definite integral is positive

How can I prove this: if $x \ge 0, $ then $$\int_{0}^{x} \frac{\sin(t)}{t+1} \, dt \ge 0 $$ I attempted this problem using monotony of integral, but I didn't get anything really useful.
0
votes
1answer
24 views

Finding upper bounds for $e^x$ in limit exercises

I'm trying to prove by definition $$\lim_{(x,y)\to(0,1)} y e^x = 1$$ How can I find upper bounds for $| ye^x - 1 |$? I know that $| ye^x - 1 | = | y(e^x - 1) + (y - 1) |$ But I never know how to get ...
0
votes
2answers
74 views

Values of $\int_{0}^{1}{\frac{dt}{1+t^n}}$

May be this question has already been asked here. I’m looking for differents methods for handling this integral. Edit: I am looking for a closed form. Any suggestion or method is welcome. Initially ...
0
votes
0answers
46 views

$f:[0,1]\rightarrow\Bbb R$ defined by $f(x)=0$ if $x$ is rational and $f(x)=2$ if $x$ is irrational.

Find the Lebesgue integral of the following functions $f:[0,1]\rightarrow \Bbb R$ defined by $f(x)=0$ if $x$ is rational and $f(x)=2$ if $x$ is irrational. I know that $\int_{[a,b]}f d\lambda=\sup ...
1
vote
0answers
36 views

Why is the factorial operation needed here?

Consider the following definition of the Dirichlet function which is $1$ at all rational points of the real line and vanishes otherwise: $$\lim_{j\to\infty}\lim_{k\to\infty} (\cos(k!\pi x))^{2j}.$$ ...
0
votes
1answer
30 views

Partial derivative of $f(u,v)$

Let $f(u,v) = c$ where $u(x,y) , v(x,y)$ are functions and $c$ is constant. Can we conclude $\frac{\partial f}{\partial v} = \frac{\partial f}{\partial u} = 0$ ? It really sounds confusing to me but I'...
0
votes
2answers
19 views

continuity of real valued functions at irrational points

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a function dfined by $$ f(x) = \begin{cases} 0,\quad x\not\in \mathbb{Q}\\ \frac{p}{p+1}, \quad x=\frac{p}{q} \end{cases} $$ Where is $f$ continuous? I know ...
0
votes
0answers
33 views

Prove that $f$ is bounded when higher derivative is bounded

Let $f:(-2019,2020) \rightarrow \mathbb R$ is $1000$ times differentiable and $f^{(1000)}$ is bounded then also $f$ is bounded. If $f:\mathbb R \rightarrow \mathbb R$ it is also right? My try: $f^{(...
0
votes
1answer
28 views

Limit as a definite integral (Riemann Sum)

I'm having a little trouble with a question that requires me to interpret a limit as a Riemann sum for an integral. However, I'm having trouble identifying which aspects of the limit correspond to the ...
6
votes
1answer
59 views

How to prove $f$ is $C^\infty$

Suppose $f:U \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ is continous and $$(x^2+y^4)f(x,y)+(f(x,y))^3=1 \: \text{for all} \: (x,y) \in U. $$ Prove $f$ is $C^\infty$. This kind of exercise ...
0
votes
0answers
20 views

How to get inequality ‎$‎0‎\leq ‎\gamma‎_g + ‎log ‎g(1)\leq ‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎$‎?

‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w&...
0
votes
1answer
25 views

$f(t) \ge 0$, $f'(t) < 0$, $f(t)$ is continuous, is it true that $\lim_{t \to \infty}f(t) = 0$?

Given a continuous function $f(t): \mathbb{R}^{+} \to \mathbb{R}$ such that its first derivative exists and is negative, namely $f'(t) < 0$. Is it true that $f(t) \to 0$ as $t \to +\infty$? In my ...
0
votes
0answers
26 views

A Hilbert measurable space is $\sigma$-finite? [on hold]

Let $\mathcal{H}$ be the Hilbert space, $\sigma(\mathcal{H})$ the $\sigma$-algebra generated by this space and $G$ a Gaussian measure defined in this measurable space. The, can I say that $(\mathcal{H}...
0
votes
1answer
27 views

Closedness of the Set of Continuous and Increasing functions

Let $ A=\left\{ f\in C\left[ 0,1\right] |\text{ }f\text{ is strictly increasing and }f\left( 0\right) =0\text{ and }f\left( 1\right) =1\right\} $, where $ C\left[ 0,1\right]$ is the set of continuous ...
0
votes
2answers
24 views

Differentiate under expectation sign.

Let $f(x)$ be the probability density function of a $\text{Uniform}(0,1)$ distribution. Let $Z\sim\text{Normal}(0,1)$ and $g(x)=E[f(x+Z)]$. If one plots $g(x)$ in a computer, one observes like an ...
1
vote
0answers
24 views

Measure and function

Any help with this question? Let $X$ be an nonempty set and $f\colon X\to[0,+\infty]$ a function. Defines $$\sum_{x\in E} f(x):=\sup\left\lbrace\sum_{x\in F}f(x);F\subset E\ is\ finite\right\rbrace....
-2
votes
0answers
52 views

Intro to math analysis

Can anyone give me some hint on this question? I don’t know how to start.
0
votes
0answers
9 views

A question about exterior measure on R

$m^*(E)=q>0$,for any $c\in (0,q)$,there exist $E_0\subset E$,such that $m^*(E_0)=c$ $$m^*(E)=inf\{mG|E\subset G ,\text{G is open set}\}$$ I think if I can find a set A$\subset E$,and $m^*(E-A)=c$,...
1
vote
1answer
23 views

Bounded sequences in $L^2(0,\infty)$ [duplicate]

Let $f_n$ be a bounded sequence in $L^2(0,\infty)$. Does this sequence have a weakly convergent sub-sequence $f_{n_k}$ in the following sense: $$\int_0^\infty \phi(t)(f_{n_k}(t)-f_0(t))dt\to 0, \quad \...
0
votes
0answers
27 views

For which $a,b,c \in \mathbb R$ series is convergent?

For which $a,b,c \in \mathbb R$ series $$\sum_{n=1}^{+\infty} (\arctan (\frac{1}{\sqrt[3]{n}}) + \frac{a}{n}+\frac{b}{n^c} +\frac{c}{n^2})$$ is convergent? My try: $$(\frac{1}{\sqrt[3]{n}}) + \frac{a}...
0
votes
1answer
10 views

Constant for a Poincaré-Type Inequality

Given a bounded oben set $\Omega\subset\mathbb{R}^n$ and an arbitrary function $u\in C^2$. Is it possible to show $$ \|u\|_{L^2(\Omega)}^2 \leq C \|\nabla u\|^2_{L^2(\Omega)}, $$ for some $C\in\mathbb{...
2
votes
0answers
42 views

Definitions of Sobolev Spaces - are they the same?

I have read two definitions of Sobolev spaces. Definition 1: We let $\lambda$ denote $\lambda^s(\xi)=(1+|\xi|^2)^\frac{s}{2}$ for $s \in \Bbb R$, $\xi \in \Bbb R^n$. We say that $u \in H^s$, if $u ...
1
vote
2answers
85 views

$\lim_{n\to\infty}\int_0^1 (f(x))^n d(x) = 0$

Let $f:[0,1]\to \mathbb{R}$ be a strictly increasing function such that $f(0)=0$ and $f(1)=1$. How to prove that $$ \lim_{n \to \infty} \int_0^1 (f(x))^n d(x)=0.$$
1
vote
1answer
25 views

Book about summable families

after studying summable series, I know that there exists the concept of summable families. There is any book where I can find the proof of the basic properties of summable families? Thanks.
-1
votes
1answer
33 views

Improper integral 1

integration $\displaystyle\frac{x\tan^{-1}{x}}{\sqrt[3]{1+x^4}}$ on the interval 0 to infinity. Is the above improper integral convergent? Plz give the solution