Theoretical foundations of calculus: 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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Equivalent Criteria for convergence of a sequence

My textbook of Metric Spaces describes the following equivalent criteria for convergence : $(i)~ \bigcap \{ \overline {x_n~|~n \in S}~|~S \subseteq \mathbb N, S$ infinte $\} = \{x\} $ $(ii)~dist~(z, ...
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8 views

The continuous embedding of weight L^1$ space.

Take $\omega_0$: $\mathbb R^N\to \mathbb R^+$ such that $\omega$ l.s.c.and $\omega_0\geq 1$ and satisfies $$ \frac{1}{|{B}|}\int_{B(x,r)} \omega_0(y)\,dy\leq C\omega_0(x) \tag 1 $$ for any Ball ...
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1answer
31 views

A series of continous functions is continuous.

Given a continuous function $f_0: [0,1] \rightarrow \mathbb{R}$, define $$f_n(x) = \int^x_0 f_{n-1}(t) dt, x \in [0,1]$$ for $n=1,2,3,...$ . For each $x \in [0,1]$, show that ...
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10 views

Continuous dependence on initial conditions for second order eigenvalue problem

Consider the Schrödinger eigenvalue problem in one dimension $$\phi'' - V\phi + \mu \phi = 0$$ on $[0,a]$ with boundary $\phi(a) = c$. Suppose that I already have the existence of ...
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18 views

Open and closed sets in an arbitrary metric space example

I have the following point lifted directly from my notes on open and closed sets in arbitrary metric spaces: "As an example of a set that is not open, consider $I=[0,1]\subset\mathbb R$, with the ...
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15 views

An inequality involving supremum and integral 2

In the following post An inequality involving supremum and integral, there was discussed the inequality $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive ...
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22 views

Multiplication operator on $L^p$ for $1<p<\infty $

If $(X,M,\mu) $ is an arbitrary measure space and $\phi :X\rightarrow \mathbb{C}$ measurable, such that $\phi f\in L^1, \forall f\in L^p$ for $1<p<\infty $,define $T:L^p\rightarrow L^1$ by ...
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11 views

Please gauge my understanding of this-A proof from GH hardy's Book

What i did to understand this was i took took a unit length $A_{1}A_{2}$. And BC=0.4 such that it lies inside it. Taking k=3. The line segment $A_{1}A_{2}$ IS divided into 3 equal parts say $p_1$, ...
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1answer
29 views

Real Analysis: function achieves minimum value

Problem Suppose that $A\subset \mathbb{R}^n$ is closed and $B\subset \mathbb{R}^n$ is compact and that no point of $\mathbb{R}^n$ lies in both $A$ and $B$, and neither $A$ nor $B$ are empty. For each ...
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1answer
30 views

An stonger form of the existence of a smooth Urysohn function on $\mathbb R^n$

I proved the following form of the existence of a smooth Urysohn function:: proposition: For any compact set $K\subset\mathbb R^n$ and any open set $U\subset\mathbb R^n$ where $K\subset U$, there is ...
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1answer
21 views

Existence of probability measure, $\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)$

Problem statement: Let $x_j$ be a sequence in $[0,1]$. Consider $$\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)\hspace 2cm (\star).$$ If ($\star$) exists for all $f\in C[0,1]$ then there is a ...
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4answers
54 views

Show that $x^2-\cos(x)$ has two roots in the real numbers

I'm stuck in a problem. I have to show that the function $f:\;\mathbb{R}\rightarrow\mathbb{R}$, defined by $f(x)=x^2-\cos(x)$ has exaclty two roots in the real numbers. It's also specified, as a ...
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1answer
16 views

Calculate the measure of a measurable set under nonlinear mapping.

It is known that: If $\cal{A} \subset \mathbb{R}^n$ is Lebesgue measurable, and $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear mapping, then $L(\cal{A})$ is Lebesgue measurable and ...
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6 views

Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, ...
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1answer
29 views

Every open set S in $R^n$ can be expressed in one and only one way as a countable disjoint union of open connected sets.

In the following proof in my book their is something I don't understand I will present the proof then present the statement which I don't quite agree with. I want to also ask about some stuff I don't ...
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30 views

“Continuity” of minimum of a function

Following is a question I encounter while reading a topic on 'large deviations'. I abstract the problem here. I don't know whether the title is apt! Let $\mathbb{A}=\{a_1,\dots,a_d\}$ be a finite ...
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38 views

Is this a correct proof of the inverse function theorem?

This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.). I have found an alternate proof of one part of the theorem but I ...
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2answers
72 views

Can this integral diverge?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be a continuous non-increasing bounded function, and suppose $$\limsup_{x\to +\infty}\frac{f(x)}{f(2x)} = +\infty. $$ Can $\int_0^{+\infty} f(x)dx$ diverges? ...
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1answer
14 views

Prove $\sup_x|u_x(x,t)|\le Ct^{-\frac{3}{4}}\|f\|_2$ for all $t>0$.

Let $u$ be a bounded solution to the heat equation $u_t-u_{xx}=0$ in $-\infty<x<\infty,t>0$ with $u(x,0)=f(x)$ with $f\in L^2(\Bbb R)$. Prove that there is a constant $C>0$, independent ...
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2answers
20 views

A proof of a known identity using Fourier series

The exercise Let $f$ be a $2\pi$ periodical function defined as $f(x)=\cos ax, \; |x|\leq \pi, \; a \notin \mathbb{Z}$. Expand $f$ in a Fourier series and prove that: $$\pi \cot \pi a = ...
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1answer
33 views

Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
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25 views

Natural extension of a discontinuous function

Let $u : \mathbb{R} \to \mathbb{R}$ be the right continuous version of the Heaviside step function. What does the natural extension $u^*$ of $u$ to the set $\mathbb{R}^*$ of the hyperreals look like? ...
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1answer
28 views

Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$ [on hold]

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr ...
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1answer
31 views

What can be said about the map?

$f\colon \mathbb R^{2}\rightarrow \mathbb R$ is a continuous map that assumes $0$ for only finitely many points. Then which one is true A. either $f(x)\le 0$ for all $x$ or $f(x)\ge ...
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1answer
34 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
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1answer
28 views

Proof that outer measure of interval equals length, why use Heine-Borel?

Define by $$ m^*(A) := \inf\left\{ \sum_i |I| : A \subseteq \bigcup_i I_i \right\} $$ the outer measure of some set $A \subseteq \mathbb R$. Then we have $m^*(I) = |I|$ for each interval (open, ...
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3answers
765 views

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano's Theorem, which, honestly, I find a bit ...
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1answer
16 views

Reiterating the piecewise-and-uniform-limit operation

Probably a hopeless question, but: Let $C$ be the class of constant functions $f$: $[a, b]\longrightarrow\mathbb{R}$. Let $\mathcal{U}(\mathcal{P}(C))$ denote the class of uniform limits of piecewise ...
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2answers
25 views

Problem with understanding the application of the Intermediate Value Theorem in the proof of the Mean Value Theorem for Integrals

I am struggling to understand the last parts of this proof because I know that the IVT states that on the interval $[a,b]$ of $f$, where it is continuous, there exists a value $L$ between $f(a)$ and ...
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1answer
51 views

The existence of a smooth functions taking values $0$ and $1$ on two given closed sets

In theorem 5.1 on page 39 Boothby's book(An introduction to Differentiable Manifolds By William M. Boothby), he prove that: Let $F\subset \mathbb R^n$ be a closed set and $K\subset \mathbb R^n$ ...
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2answers
33 views

$\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ for $\Delta u=1$?

Let $U\subset\mathbb R^n$ be open and bounded. Let $\Delta u=1$ in $U$. Can you follow $\sup_{x\in B_r(z)}|\nabla u(x)|\geq cr$ provided $B_r(z)\subset\subset U$? I thought about using the ...
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2answers
27 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
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1answer
44 views

Function $f \geq 0$ a.e.

Suppose $f : \mathbb{R}^d \to \mathbb{R} $ is integrable (with respect to measure $\mu$) and for every measurable set $E \subset \mathbb{R}$ we have $\int_{E} f d \mu \geq 0$. Prove that $f \geq ...
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1answer
34 views

Surface integral on unit circle

Let $S$ be the unit sphere in $\mathbb{R}^3$ and write $F(x)=\nabla V(x)$ where $V(x)=1/|x|$ Evaluate $$\iint_S F\cdot n dS$$ Without using divergence theorem, we can evaluate it straightforwardly, ...
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20 views

Show that a collection of finite unions of sets of the form $(a,b]\cap \mathbb{Q}$ is an algebra

The following is a question from Folland's Real Analysis: Modern Techniques and their Applications. (Question 23 page 32) Let $\mathcal{A}$ be the collection of finite unions of sets of the form ...
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1answer
58 views

“Mean value like” problem.

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad ...
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1answer
20 views

Improper integral parametrised in complex variable: when is it holomorphic?

Suppose we are considering the following integral: $$ I(s) = \int_1^\infty t^{s-1}e^{-t\lambda}\;dt $$ where $s \in \mathbb{C}$ and $\lambda > 0$ is a fixed constant. I would like to know when this ...
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3answers
53 views

Find x for which $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges.

For what values of $x$ the infinite series $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges? Please help me out. Using ratio test $\large{\lim_{n\to ...
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Can we find real values of $x$ such that $\lim_{n\to \infty}c^n\prod_{k= 1}^n| a_k x-1|=0$?

Let $\{a_n\}$ be a sequence such that the sequence of Cesaro means is convergent i.e. $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n a_k$$ exists. Let $c>1$ be a given number and I am trying to find ...
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2answers
66 views

Is it possible to find $a$, $b$ such that $(a,b] \cap \mathbb{Q} = \emptyset$?

Is it possible to find $a$, $b$ $\in \mathbb{R}$ satisfying $- \infty \leq a < b \leq \infty$ such that $(a,b] \cap \mathbb{Q} = \emptyset$? I need this result for a proof I am doing...
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1answer
39 views

How to calculate $lim_{n\rightarrow \infty} \dfrac{n+\sin 2n}{\sqrt{n}-\cos n }$ [on hold]

I have to solve this problem. $$lim_{n\rightarrow \infty} \dfrac{n+\sin 2n}{\sqrt{n}-\cos n }$$
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1answer
34 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
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0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
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2answers
101 views

Is $(a,a]=\{\emptyset\}$?

Let $a \in \mathbb{R}$, and consider the half open interval $(a,a]$. Is it correct to write this half open interval as $(a,a]=\{\emptyset \}$? Or $(a,a]=\{a \}$?
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2answers
23 views

Proof structure validity: assume (a), (b), show (c). Then Permute.

I am given a collection of sets $\mathcal{E}$ and am trying to prove it is an elementary family. To show $\mathcal{E}$ is an elementary family I must show that it satisfies the following properties: ...
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2answers
44 views

Real analysis using some key constraints

Let $\alpha>1$ and $M \geq 0$. Suppose $f:\mathbb{R}→\mathbb{R}$ satisfies $$|f(x)-f(y)| \leq M|x-y|^\alpha$$ for all $x,y\in\mathbb R$. Prove that $f$ is a constant function. I tried taking ...
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1answer
41 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
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0answers
22 views

Conditions on $f$ to have $ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $ finite?

Suppose that $f$ is a $\mathcal{C}^\infty$ function. $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $$ Which are the conditions on $f$ that makes this integral finite ? ...
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2answers
62 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...
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0answers
32 views

Why do we need a Borel function in order to use this lemma?

Im trying to understand a proof for differentiably a.e for functions $F$ given by $$F(x)= \int_{-\infty}^{x}f\ \mathsf dt$$ for $f$ Lebesgue measurable and $L^{1}$. He defines a finite Borel measure ...