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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Uniform convergence of Lipschitz functions to characteristic function of a compact set

Consider $(X,d)$ a metric space and $K \subseteq X$ a compact subset. I am trying to build a sequence of Lipschitz functions $f_n : X \to \mathbb R$ s.t. $f_n \to \chi_K$ uniformly. If we try to ...
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0answers
11 views

Is there any solution manual to Halmos' Measure Theory?

I've spent some time on Halmos' Measure Theory and must upvote such a good book. I want to solve most exercises in this book. I'm not sure whether there is a solution manual or instructor manual that ...
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0answers
12 views

measure implies the existence of limit function

I am reading a book and I don't quite understand some of the statement. It says "$\{u_n\}$ is a Cauchy sequence in the space $L^2(\mathbb{R}^d;|\xi|^{2s}d\xi)$. Because $|\xi|^{2s}d\xi$ is a measure ...
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1answer
10 views

Limit of sequence partially applied to a function

Let $(a_n)_{n\in\mathbb N}$, $(b_n)_{n\in\mathbb N}$, $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ be real-valued sequences and $f:\mathbb R\to \mathbb R$ monotone with ...
1
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1answer
14 views

Boundedness of smooth functions approximating an Lp function

We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f ...
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1answer
18 views

Convergence of a Sequence Involving arctan - Is my solution correct?

Here's my question: Let $(a_{n})$ be a sequence where $(a_{1}) > 0$, defined as: $$(a_{n+1})=\arctan*(a_{n})$$ for all $n$. Prove that $a_{n}$ has a limit $L$ and calculate it. Solution: ...
4
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1answer
60 views

An elementary proof for a bound on $x \log x$

During one of our information theory classes, the Professor used the following bound to prove a result. For any $x,y \in (0,1)$, $x \neq y$, show that $$|x \log(x) - y \log(y) | \leq |x-y|\log ...
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2answers
21 views

Finding the intervals where $f(x)=\frac{1}{|x-2|}-x$ is monotonous

Given $$f(x)=\frac{1}{|x-2|}-x.$$ I am interested in finding the intervals in $\mathbb{R}$ in which the function is monotonically increasing or decreasing. Usually I would take $f'(x)>0$ for the ...
2
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1answer
29 views

Problem 20 chapter 3 from baby Rudin

Suppose $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subseqeunce $\{p_{n_i}\}$ converges to a point $p\in X$. Prove that the full sequence $\{p_n\}$ converges to $p$. Proof: ...
8
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1answer
47 views

Will the Lebesgue integral of a real valued function always be a Riemann sum?

If we have a real valued integral that is Lebesgue integrable but not Riemann integrable, can the value of the Lebesgue integral be given by a Riemann sum by choosing appropriate points in the ...
2
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2answers
39 views

$\{a_n\} \to a$ iff $\limsup_{n \to \infty} \{a_n\} = \liminf_{n \to \infty} \{a_n\}$

It is clear that if $$\limsup_{n \to \infty} \{a_n\} = \liminf_{\to \infty} \{a_n\},$$ then $\{a_n\} \to a$, since we can just squeeze the terms in the middle. I understand that to prove the ...
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0answers
67 views

Why are definitions written as 'if-then' statements instead of 'if-and-only-if' [duplicate]

An example from Rudin would be: (c) if $x + y = 0$ then $y = -x$. There may be times when one would have to use the fact that since $y = -x, x + y = 0$. While this is fairly intuitive, professors ...
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1answer
32 views

Confusion about Partial Derivative for a Function of One Variable

This question actually came up as I was reading an example in my differential equations book (Boyce & Diprima): Solve: $2x+y^2+2xyy'=0$ Define $\psi(x,y)=x^2+xy^2$ Then ...
6
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1answer
56 views

A Vitali set is non-measurable, direct proof, without using countable additivity

I am teaching a measure theory class, where we are in the process of constructing Lebesgue measure on $\mathbb{R}$ via the usual Caratheodory outer measure construction. As motivation, we began by ...
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1answer
48 views

Proving $x_ky_k\to ab $

Prove that the sequence $x_ky_k\to ab $ if $x_k\to a$ and $y_k\to b$. I wanted to try and do this with the epsilon definition but i am having a few technical issues. Proof: $$ |x_ky_k - ab| = |x_k ...
3
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1answer
35 views

Making sense of $ f(y) - f(x) = \int_{\tau = 0}^{1} \langle \nabla f( x+ \tau (y - x)), y - x \rangle d \tau $

I was wondering if anyone has a good explanation why this holds. I came across this in the page 17 of this paper (equations at the end of the page): $$ f(y) - f(x) = \int_{\tau = 0}^{1} \langle ...
0
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1answer
24 views

Is the closure of a bounded open set in $\mathbb{R}^n$ locally convex about the boundary?

The question is basically in the title. I think it is true, but I'm not sure. Can someone verify whether this is in fact true. Thanks.
2
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1answer
24 views

Exercise: signed measures, total variation.

I have this exercise: Let $\nu_1$ and $\nu_2$ be finite signed measures on $(\Omega,\mathcal{A})$. Prove that: $|\nu_1+\nu_2|\le|\nu_1|+|\nu_2|$; , that is ...
4
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0answers
39 views

Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
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3answers
55 views

Prove: set of the polynomials of degree $n$ with coefficients in $\Bbb Q $ is countable

Let $P_n =\{p(x)=a_n x^n+ a_{n-1} x^{n-1}+...+ a_1 x+a_0 |a_i \in \Bbb Q \}$ the set of the polynomials of degree $n$ with coefficients in $\Bbb Q $ Prove that $P_n$ is countable and tell why $P= ...
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1answer
34 views

Decimal expansion of $x\in [0,1]$

This is an exercise from Royden Real Analysis: Let $p$ be a natural number greater than 1, and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\{a_n\}$ of integers with $0 \leq ...
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1answer
20 views

Measurable Sets Definition

Definition: A set E is said to be measurable provided for any set $A$, $$\mathit{m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)}$$ (where $m^*$ is consider to be outer measure We define the outer measure ...
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0answers
15 views

bi-Lipschitz equivalence

I'm trying to prove this statement: For a $C^1$ manifold $M$ every point of $M$ has a neighborhood that is $(1+\varepsilon)$-bi-Lipschitz equivalent to a piece of $\mathbb{R}^n$.
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3answers
28 views

How to prove the Bolzano-Weierstrass theorem in the Euclidean space.

Theorem: Let $A$ be a bounded infinite subset of $\mathbb{R}^l$. Then it has a limit point. So this is the Euclidean version of the Bolzano-Weierstrass theorem, the thing is that I was trying ...
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2answers
47 views

Zeros of an analytic funtion

Are the zeros of a non-constant real analytic function $f$ from a finite dim, real vector space $V$ to the real numbers $\mathbb{R}$ which takes values in $[0,1]$ always a countable set? Update: Is ...
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3answers
40 views

Proving that this set is open in a metric space.

Let $A$ be a nonempty set in the metric space $(X,d)$ and, for $\epsilon>0$, define $$A_\epsilon = \{x\in X: d(x,A) < \epsilon\}$$. Then I want to prove that $A_\epsilon$ is open in $X$. So ...
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1answer
26 views

Choosing a smooth function with desirable properties

Consider a smooth function $\varphi \in C^\infty[0, 1]$, where $\varphi (1) = 0$. My question is, can we necessarily choose another function $\psi \in C^\infty[0, 1]$, such that $\psi \geq 0, \psi(1) ...
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0answers
22 views

Example of absolutely continuous function whose integral can't be computed exactly

I'm reading up on AC functions (I need the background of such functions for my BSc degree thesis) but I only come across theorems, lemmas etc. I have two questions: I've read conflicting things in ...
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0answers
18 views

The (matrix) definition of a positive-definite function

In the definition of a positive-definite function: https://en.wikipedia.org/wiki/Positive-definite_function Why are the elements of the n$\times$n matrix chosen as $f(x_i-x_j)$ for i, j = 1,...,n? ...
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1answer
32 views

Find lower and upper bound of $f: \{x \in \mathbb{R}^3 : x_{1}^2 + 2x_{2}^2 + 3x_3^2 \le 6 \} \to \mathbb{R}$

$f$ is given by the formula : $f(x) = 2x_1 + 4x_2 - 6x_3$ Since the domain of $f$ is a bounded and closed set, $f(x)$ does have upper and lower bounds, either in the interior of its domain or on the ...
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5answers
75 views

Which of the following condition ensure that the function $f:R^n\to R$ is continuous?

I encountered an interesting problem in my Economics class about continuity. Which of the following conditions on the function $f:\mathbb R^n\to \mathbb R$ ensures that the function $f$ is ...
1
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1answer
10 views

Diophantine approximation with integer vectors

I would like to determine whether or not there exists ${\beta > 0}$ and ${\gamma \geq 2 }$ such that ${ \forall (m_{1},m_{2}) \in \mathbb{Z}^{2} \setminus (0,0) }$, one has the inequality $$ ...
0
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1answer
24 views

Which are the good books,resources,extensive question banks to learn real analysis,calculus

Which are the good books,resources,extensive question banks to learn real analysis,calculus(indefinite,definite,area under curves),differential equations for IIT plus plus level.Foreign authors are ...
3
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1answer
57 views

On the integration of a Lebesgue measurable function

Consider a function $f$ defined as $f:[0,2\pi]\to \mathbb{R}$ such that $\begin{equation} f(x)=\inf_{n\in \mathcal{N}} \sin^2 (2^n x) \end {equation}$ Is possible to give a decent bound of ...
2
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1answer
39 views

Using epsilon and delta to compute a derivative

Let $a>1$, let $x\in\mathbb{Q}$, and define $f(x)=x^a$. I am interesting in computing $f'(0)$ if it exists. I claim that $f'(0)=0$. Attempt: Let $\epsilon > 0$. Suppose $0 < \lvert x-0 ...
0
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2answers
23 views

Interior points are limit points in $\mathbb{R}$?

I have read another question, and know that interior points are not limit points in general topology space. But when we talk about any subset $\mathbb{A}$ of $\mathbb{R}$, can I say that ...
5
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2answers
87 views

Condition implying rationality of $u^n+v^n$

$Given :\ u+v \ is \ rational, \ u^2 + v^2 =1 \ , prove \ v^n + u^n \ is \ rational$. What I have done so far is proving that $uv$ is rational by expanding $(u+v)^2$. I expanded $(u+v)^n$ using ...
2
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3answers
111 views

Inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$

I am trying to find a proof of the following inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$ and ...
4
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2answers
63 views

Showing that $(f_1,f_2,\dots,f_m)$ is a measurable function from $(\mathbb R^m,\mathcal B(\mathbb R^m))$ into itself

If $\{f_i, 1 \le i \le m\}$ is a set of real valued Borel functions on $\mathbb R$, how to show a vector of functions, $(f_1, f_2,..., f_m)$ is a measurable mapping from $(\mathbb R^m, ...
2
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3answers
51 views

Boundary of a bounded open set in $\mathbb{R}^2$

Does the boundary of a bounded open set in $\mathbb{R}^2$ necessarily have infinite points? How do we prove that, or is there a counterexample? It seems true to me, but I haven't been able to find a ...
3
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1answer
39 views

If $\lambda=$ measure of a set and all $G_k$'s are open sets, then : $\lambda ( \cup_{k=1}^{\infty} G_k ) \le \sum _{k=1}^{\infty}\lambda ( G_k)$

I just started reading the book Lebesgue Integration on Euclidean Spaces by Frank jones, in which the author gives a result and it's proof as : the If $\lambda$ denotes the measure of a set and all ...
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1answer
33 views

if $a,b,c,d \in \mathbb R$ such that $a < b$ and $c < d$, then prove that $[a,b]$ is equivalent to $[c,d]$. [on hold]

What am I supposed to do? I'm relearning cardinality of sets, Archimedean property, infimum and supremum...
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0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
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1answer
65 views

Folland, Real Analysis Theorem 1.19

Theorem: If $E\subset\mathbb{R}$, the following are equivalent a.) $E\in M_\mu$ b.) $E = V\setminus N_1$ where $V$ is a $G_\delta$ set and $\mu(N_1) = 0$ c.) $E = H\cup N_2$ where $H$ is a ...
0
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1answer
22 views

Approximation of continuous functions by Bernstein polynomials

Recently a professor show me the following heuristic to provide approximations of continuous functions by polynomials: Let $P_n(x) = \sum_{k=0}^{n} {n \choose k} f(\frac{k}{n}) x^k (1-x)^{n-k}$. ...
0
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1answer
31 views

Finding the adherent points of $A=\left\{\left(1/n,1/m\right)|n,m\in\mathbb{N}\right\}$

The obvious adherent point is $(0,0),$ then I thought about fixing a point for each component and finding the adherent points on each line, but it leaves a mess. Doing it "my way" would lead to find ...
1
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1answer
24 views

Trying to construct a specific function

I am trying to construct a function $f$ with the following property: $\mathbf{N}$ is the set of natural numbers without 0. Show that $\forall \epsilon>0: \forall a,b \in \mathbf{N}: a < b: ...
1
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2answers
62 views

Convergence of a sequence of convolutions

Let $(a_n)$ be a sequence of real numbers such that $$ a_0>a_1>\cdots>0 $$ and $M:=\sum_{n=0}^\infty a_n<+\infty$. Denote $$ g_n=\frac{1}{a_n}\cdot 1_{[0,a_n]} $$ and define ...
2
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3answers
106 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
0
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1answer
33 views

When is it ok to use a sequential limit in place of a continuous limit?

I am working through some Lebesgue integral problems, and I've come across a few instances where I would like to use the dominated/monotone convergence theorems, but the limit is continuous, and I'm ...