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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.

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0answers
27 views

An increasing real function is integrable.

If $f:\mathbb{R}\to \mathbb{R}$ is an increasing function, then $f$ is integrable I saw many times with domain $[a,b]$ that the set of discontinuities of $f$ has measure zero. The secret of the proof ...
1
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1answer
45 views

Is the function $f: \mathbb{Q}\to \mathbb{R}$ given by $f(r)=\frac{p}{10^q}$ one-one and onto?

Is the function $f: \mathbb{Q}\to \mathbb{R}$ given by $f(0)=0, ~$$f(r)=\frac{p}{10^q}$ where $p\in \mathbb{Z}$ and $q\in \mathbb{N}$, $(p,q)=1$, one-one and onto? Since, for $\sqrt{2}\in \mathbb{R}$ ...
1
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2answers
33 views

Multiply a Known Convergent Sequence by a Bounded Sequence

I have been tasked with proving the following: $a_n$ is bounded but not necessarily convergent assume that $\lim{b_n} \to 0$ show $\lim{a_n b_n} \to 0$. I started my proof listing what I ...
2
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1answer
52 views

Pathological, continuous functions

Today in my introduction to measure theory course, the professor said that often when we think of continuity, what we're actually thinking about is smooth functions. We've studied the Cantor set and ...
2
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1answer
53 views

Proving Fatou's lemma from the DCT

I was recently told that the "Big Three" convergence theorems for the Lebesgue integral (Fatou, dominated, and monotone) are equivalent. I'm trying to show this directly by writing six proofs instead ...
3
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2answers
65 views

Understanding the proof $C(X)$ is complete

Prove $(C_X,||.||)$ ,where $||.||$ is the maximum norm and X is compact, is complete. The following proof was given. It is the one I am striving to understand: Let $(f_n)$ be a Cauchy sequence: $\...
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1answer
27 views

Preimage of $\partial\Omega$ under a continuous function

Suppose $\Omega\subset \mathbb{R}^n$ is open and bounded, and let $\partial\Omega$ the boundary of $\Omega$. Fix $x \in \overline{\Omega}=\Omega \cup \partial\Omega$ and let $y:[0,+\infty)\to \mathbb{...
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3answers
65 views

How to prove that $f(x) =1/x$ is unbounded in $(0,1)$?

Let $f(x) = 1/x$ for all $x\in (0,1)$ By assuming $f(x)$ is bounded or in any other way but without using limits I assumed that $f$ is bounded above, then there is $M>0$ s.t $f(x) \leq M$ for all ...
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1answer
39 views

Why can't we prove that union of infinite no of countable sets is also countable by induction?

If $A1, A2,...., A_m$ are each countable sets then $A1\cup A2\cup.... \cup A_m$ is countable. Why can't we use induction to prove that if $A_n $ is countable for all n then $\bigcup_{n=1}^\infty A_n$...
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2answers
166 views
+50

$f(x)/x \to l$ and $f''(x) = O(1/x)$

$f \in C^2(\mathbb{R}, \mathbb{R})$ such that $f(x)/x \to l \in \mathbb{R}$ as $x \to + \infty$, and such that $f''(x) = O(1/x)$ at $+\infty$. Find : $$\lim_{x \to +\infty} f'(x)$$ Some thoughts : ...
2
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1answer
21 views

Koopman-von Neumann Decomposition Properties

Sorry if the title of this question is vague--I'm open to suggestions. For this question, we're working in a probability space $(X, \mathcal{M}, \mu)$. In a proof of "ergodic Roth's theorem" given ...
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1answer
51 views
+150

Questions on the real bump function and conjuction of smooth functions

Before I ask you my question which I will mark in bold I will tell you what I already gathered so far. In a previous result I have showed that the bumpfunction is smooth. The bumpfunction is defined ...
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1answer
20 views

An inequality about compact functions and lsc functions

I'm reading a proof and it contains the following inequality: Suppose $u: E \rightarrow [0,\infty]$ is lower semicontinuous and let $u_t$ be a sequence of Lipschitz functions approaching $u$ from ...
1
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1answer
38 views

Probability estimation question.

Consider a random walk with initial point zero. So we have $S_{n} = X_{1} + \dots + X_n$. And we have two fixed numbers $b > 0 > a$. Now we want to show that $(*) = \operatorname{P}(\{\forall n :...
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2answers
38 views

closure of the set $S=\{(x,y)|x\in(0,1)\cap\mathbb{Q};y=\sin(\frac1x)\}$

I am trying to determine the closure of the set $S=\{(x,y)|x\in(0,1)\cap\mathbb{Q};y=\sin(\frac1x)\}$. I don't know where to start. If we define a sequence $(x_n,y_n)=(x_n,\sin(\frac{1}{x_n})),x\in(...
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4answers
93 views

Is there any way to show that $\sum_{n=2}^{\infty}\frac{1}{n^2-1}$ converges without integral test?

In my real analysis class, we are trying to show that $$\left(\frac{2 \times 2}{1 \times 3}\right)\left(\frac{4 \times 4}{3 \times 5}\right) \left(\frac{6 \times 6}{5 \times 7}\right)...$$ converges. ...
7
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5answers
668 views

Stuck at proving whether the sequence is convergent or not

I have been trying to determine whether the following sequence is convergent or not. This is what I got: Exercise 1: Find the $\min,\max,\sup,\inf, \liminf,\limsup$ and determine whether the ...
2
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2answers
75 views

Nice power sum inequality $ab(a-b)\leq a^{ab}-b^{ab}$ for $a+b=2$

Let $a\geq b>0$ such that $a+b=2$ then we have : $$ab(a-b)\leq a^{ab}-b^{ab}$$ My try : We study the function $f(x)$ on $[0;1]$ such that : $$f(x)=(2-x)^{((2-x)x)}-x^{((2-x)(x))}-(2-x)x(2-2x)$$ ...
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1answer
25 views

Totality in Hilbert space used to show an inequality

Let $e_k\in L_2(a,b), k=1,2,3,...$ be an orthonormal sequence in $L_2(a,b)$. I want to show if it is total then the following holds for all $x\in (a,b)$: $\sum_{k=1}^{\infty}|\int_a^xe_k(t)dt|^2=x-a$. ...
2
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0answers
58 views

Sobolev Spaces subsets of each other?

I have recently started working with Sobolev Spaces and I wanted to ask the following: Let $1 \leq p \leq \infty, \Omega \subset \mathbb{R}^d$. Does $ W^{n,p}(\Omega) \subset W^{1,p}(\Omega)$ hold? ...
1
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2answers
56 views

Sequence converging towards Euler-Mascheroni constant

I have been trying to find the limit of the following sequence: $\lim_{n\to \infty}(\frac{1}{2n+1} + \frac{1}{2n+2}+...+\frac{1}{8n+1})$ Here is my attempt with the Euler-Mascheroni constant: $(\...
1
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1answer
24 views

Can someone explain how the Cauchy's mean value theorem applies here

How isnt this a exeption to the Cauchy's mean value theorem? $f(x)=x^2+x;g(x)=x^3;x∈[−1,1]$ Because $g'(0)=0$ which is a contradiction to the third condition that: $g'(x)\neq 0 ;\forall x∈(−1,1)$
2
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4answers
90 views

A solution for $x^x=2$

I just stumbled upon this question and just can't figure out a way to prove it. Show that the equation $x^x=2$ has a solution in $\mathbb{R}^+$. I'm just curious... can someone show me, how to ...
0
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2answers
66 views

$f(x)=\sum\limits_{n=1}^{\infty}{\frac{x^{n-1}}{n}}$,Prove that $f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$

$$f(x)=\sum_{n=1}^{\infty}{\frac{x^{n-1}}{n}},$$ Prove that $f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$ In my mind though,I think that this is related to Basel problem$\left(\sum\limits_{n=...
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3answers
24 views

Confusion about relationship between continuous and real analytic functions

By Weierstraß aprroximation theorem, every continuous function can be $\epsilon$-approximated by a polynomial, i.e., if $f:[a,b]\to\mathbb{R}$ is a continuous function, then for all $\epsilon>0$ ...
2
votes
2answers
36 views

Check if the sequence is bounded?

Before down-voting my question, bare in mind that this is my first post and question here and if you can help me to improve the quality, I'd be really thankful. I am struggling with how to check if ...
3
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1answer
35 views

Proving the connection between the Binomial Theorem and the product rule for derivatives

Let $a(x)$ and $b(x)$ be smooth functions, i.e they are infinitely times differentiable. I have made the assumption that the derivative for the function $$f(x)= (a\cdot b)(x)$$ can be given by $$...
2
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1answer
55 views

Maximizing the value of an integral

Let $f \colon \mathbb R^N \to \mathbb R$ be a measurable, bounded function. Let $$ \mathcal A := \left\{ g \colon \mathbb R \to [0,+\infty): g \text{ is measurable and} \int_\mathbb R g =1\right\}. $$...
3
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1answer
61 views

Prove that integral of infinite sum is in $L^1$

The function $g$ is a composition of functions as follows ($\chi_A$ is the characteristic function of the set $A$): $$f(x) = \frac{1}{\sqrt{x}} \chi_{[0,1]}(x)$$ For an enumeration of the rationals $...
3
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3answers
59 views

A question about estimating an area of overlapping circles

I need to show $$\text{Area}((B-B') \cup (B'-B))\leq 4R\delta$$ where B and B' are circles with the same radius R, B is centered at the origin and B' is centered at $(\delta,0)$. Since $B-B'$ and $B'-...
3
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3answers
68 views

Why $\max\{f,g\}$ is continuous if $f$ and $g$ are continuous?

Let $f,g:\mathbb{R}\to \mathbb{R}$ be continuous functions. I need to prove that $x\mapsto \max\{f(x),g(x)\}$ is continuous without using the fact that $\max\{a,b\}=\frac{a+b+|a-b|}{2}$. Let $\...
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0answers
18 views

Using squeeze theorem to prove that modified zero function is Riemann integrable [duplicate]

$\text{(Riemann Integrability) Squeeze theorem :}$ Let $f: [a,b]\rightarrow \mathbb{R}$. Then $f\in \mathcal{R}[a,b]$ if and only if for every $\epsilon>0$ there exists functions $\alpha_\epsilon$ ...
0
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1answer
32 views

Estimate for $\|a^2-b^2\|_{C^2(\Omega)}$ given that $\|a-b\|_{C^2(\Omega)} \leq c$ [on hold]

Given $$\|a-b\|_{C^2(\Omega)}\leq c$$ for some $a,b\in C^2(\Omega)$. Where $\Omega \subset \mathbb{R}^n$ is bounded and $c\in \mathbb{R}$. I need to show that $$ \|a^2-b^2\|_{C^2(\Omega)}\leq \tilde{...
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0answers
42 views

Is there $\int_ {0} ^ {1} g\,dx$?

Definition: A partition $ P $ of $ [a, b] $ is a finite set $ \{x_{0}, ..., x_ {n} \} \subseteq {[a, b]} $ such that $ a = x_ {0} <x_ {1} <... <x_ {n-1} <x_ {n} = b $. The norm of a $ P $ ...
2
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1answer
49 views

Compute $\int_{0}^{\frac{1}{2}} \frac{\arctan x - \sin x}{x}\, dx $ with an error less than $10^{-2}$

I have used the Taylor expansions of the functions arctan and sin and the fact that power series uniformly converge in $[0,1/2]$ so I obtained: $$\sum \biggl(\frac{1}{(2n+1)!}-{\frac{1}{2n+1}}\...
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1answer
18 views

The image of totally disconnected compacta under Lipschitz maps

The Alexandroff-Hausdorff theorem tells that every compact metric space is the continuous image of the Cantor set. I am wondering whether the image of a compact totally disconnected subset in $\...
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1answer
25 views

About the double integral and fubini's theorem

Let $f:[0,1]\times [0,1]\to\mathbb{R}$ such that $f(x,y)=1 $ if $x= \frac{m}{n} ,y= \frac{q}{n}, (m,n)=(q,n)=1 $ otherwise $f(x,y) =0$ ,Now which of following options is true ? I think because $f(x,...
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2answers
63 views

Suppose that $A$ has Jordan content null. Show that $\partial A$ has Jordan content null.

Denote the Jordan content $\left(\displaystyle \int_{R}\chi_{A}(x)\mathrm{d}x\right)$ of $A$ by $c(A)$ and the $m(A)$ the Lebesgue measure of $A$. (a) Suppose that $c(A) = 0$. Show that $c(\...
4
votes
1answer
30 views

Exercise in Holder's inequality

The following is a problem from Royden and Fitzpatrick's Real Analysis book. Find the values of the parameter $\lambda$ for which $$ \lim\limits_{\epsilon\rightarrow0^{+}} \frac{1}{\epsilon^\...
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1answer
45 views

Show that $f(x,y)=\left(x^4+y^4\right) \log \left(x^2+y^2\right)$ with $f(0,0)=0$ is continuous.

Show that $f(x,y)=\left(x^4+y^4\right) \log \left(x^2+y^2\right)$ with $f(0,0)=0$ is continuous. Can't seem to show that the partial $\frac{2 x \left(x^4+y^4\right)}{x^2+y^2}+4 x^3 \log \left(x^2+y^...
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1answer
30 views

Equivalent condition of a diffeomorphism having a dense orbit

Say $M$ is a manifold and $f: M \to M$ is a diffeomorphism. Assume also that, if we are given any nonempty open subsets $U$ and $V$, then there is $n \in \mathbb{Z}$ such that $f^n(U)$ intersects $V$....
1
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2answers
55 views

Prove that the derived set of $A \subseteq \Bbb R$ is closed

For $A \subseteq \Bbb R$, the derived set of $A$, denoted by $A'$, is the set of all limit points of $A$. Theorem: For $A \subseteq \Bbb R$, $A'$ is closed. Does my attempt look fine or contain ...
1
vote
1answer
60 views

Can you prove mean value theorem using IVT?

Assuming derivative is continuous, let us first start with Rolle's theorem. The result follows easily if we don't assume that the derivative is either positive or negative in the interval. But if so, ...
3
votes
3answers
121 views
+50

If $x_n$ is the unique solution to $\tan x = n x$ in $(0,\pi/2)$, then is $\lim x_n = \pi/2$? (Also, a related sequence.)

For any $n \in \mathbb N, n\geq2$, we denote the unique solution of the equation $\tan x=nx$ which is in $(0,\frac{\pi}{2})$ as $x_n$. I want to study the limit of the sequence $(x_n)$, say $\ell$. ...
4
votes
3answers
56 views

Understanding why a limit proof using another limit works

Sorry for the title, hopefully I can explain it better. I think the title is about as good as I could get in terms of description. I have a problem: Let $x_n \ge 0$ for all $ N \in \mathbb{N}$ ...
0
votes
1answer
30 views

Show that there is a sequence $(P_n)$ of partitions of $[0,1]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g\colon[0,1]\rightarrow \mathbb{R}$, be defined as $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x)=1/x$ if $x \not\in\mathbb{Q}$. Show that there exists a sequence $(P_n)$ of partitions of $[0,1]$ ...
0
votes
1answer
29 views

Reference Request for Ordinary Differential Equation problem book

I am looking for a good problem book in Ordinary Differential Equations at Graduate's level. Can someone suggest me the book for problem practice. To be precise our study revolves around the analysis ...
0
votes
0answers
15 views

$f(x,y)=1 $ if $x= \frac{m}{n} ,y= \frac{q}{n}, (m,n)=(q,n)=1 $ otherwise $f(x,y) =0$ , find double integral of f

Let $f:X\times Y\to\mathbb{R}$ such that,$X=Y=[0,1]$ and $f(x,y)=1 $ if $x= \frac{m}{n} ,y= \frac{q}{n}, (m,n)=(q,n)=1 $ otherwise $f(x,y) =0$ ,Now which of following options is true ? I think ...
1
vote
2answers
60 views

Prove or disprove: convex function $f:[a,b] \to \mathbb{R}$ attains a minimum value on $[a,b]$

Let $f:[a,b] \to \mathbb{R}$ be a convex function. Prove or disprove: $f$ attains a minimum value on $[a,b].$ Attempt. I believe the answer is yes. We know: If $f:I\to \mathbb{R}$ is convex and ...
0
votes
0answers
13 views

Show $\int_{dM} w=\int_M dw$, use $d(w|_M)$ not $w\in \Bbb A(\Bbb R^3)$ , $w|_M \in \Bbb A(M)$.

Let $w \in \Bbb A^1(\Bbb R^3)$, $w=xzdy-yzdx$, $M=\{z=f(x^2+y^2\} over $x^2+y^2 \leq \Bbb R^2$. Illustrate Stoke's theorem with $(M,w|M)$. For easier use $f(x)=x$. Show $\int_{dM} w=\int_M dw$, use $...