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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24 views

Is every differentiable function on $(0,1)$ uniformly continuous $?$

$$f:(0,1)\rightarrow [0,1]$$ is a differentiable function . Is it uniformly continuous then $?$ Now $f$ being differentiable on $(0,1)$ is continuous , that is easy. Now I could ...
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0answers
16 views

Baby Rudin Exercise 2.24

I have some difficulties solving the following exercise (Baby Rudin 2.24) Let $ X $ be a metric space in which every infinite subset has a limit point. Prove that $ X $ is separable. In order to ...
2
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4answers
26 views

Sum of two harmonic alternating series

I'm trying to solve the series $\sum_{n=1}^\infty (-1)^{n+1}\frac{2n+1}{n(n+1)}$ I've simplified it to the form $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n+1} + \sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ ...
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4answers
62 views

The sequence $x_{n+1}=ax_{n}+b $ converges to where?

$$a,b \in \mathbb R , \ 0\lt a\lt 1 . $$ Define the sequence $$x_{n+1}=ax_{n}+b \text{ for } n\ge0\ .$$ Then for a given $\ \ x_0\ \ $ , does this sequence converge? And if it does, to ...
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1answer
17 views

Existence of the partial derivatives ${\delta^{2}f}\over {\delta x \delta y}$ and ${\delta f}\over {\delta x}$

The question is can the partial double derivative ${\delta^{2}f}\over {\delta x \delta y}$ exist without the derivative ${\delta f}\over {\delta x}$ existing? I don't know , I am ...
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0answers
29 views

Does Every Periodic Function Have An Associated Differential Equation?

My question is the opposite to proving the existence of periodic solutions to ODE's. Assume that $\ f(z)$ is a periodic function over the $\mathbb{R}$ , or doubly periodic over some lattice $\Lambda$ ...
0
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1answer
24 views

Measurable sets

Show the following. E is measurable $\Longleftrightarrow$ For any $\epsilon > 0$, there exists a closed set $F \subset E$ such that $m^*(E) - m^*(F) < \epsilon$ Here is my attempt: $"\leq"$ ...
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1answer
64 views

Calculating in closed form $\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$

It's not hard to see that for powers like $1,2$, we have a nice closed form. What can be said about the cubic version, that is $$\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$$ What are ...
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1answer
34 views

Is the metric ${d(x,y)}\over {1+d(x,y)}$ complete where $d$ is the usual Euclidean metric on $\mathbb R^{2}$

Let $d(x,y)$ be the usual Euclidean metric on $\mathbb R^{2}.$ $\mathbb R^{2}$ is complete under $d(x,y)$. I have this subspace given $$[0,1]\times [0,\infty )\ \ of\ \ \mathbb ...
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0answers
15 views

Every $f\in\omega^\omega$ is bounded by the “increasing enumeration” of the intersection of a countable dense set and a dense open set in $\mathbb{R}$ [on hold]

I am studying the theorem 2.2.6 of "On the structure of the real line" of book Bartosznky-Judah. In the proof of theorem 2.2.6 the part $(4) \to (5)$ Fix a countable dense subset $X=\{q_n:n \in ...
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0answers
10 views

To show that $L(f) = \sup \{L(P;f) : P \in P^*\}$.

Let $a>0$ and let $J = [-a , a]$. Let $f: J \to \Bbb R$ be bounded and $P^*$ be the set of all partitions $P$ of J that contain $0$ and are symmetric. Show that $L(f) = \sup \{L(P;f) : P \in ...
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3answers
41 views

Proving a function is bounded

Here's my question, and a suggestion for a solution. Please let me know if I'm wrong. Prove that the function $$f(x)=\frac{\ln(x+1)}{x}$$ is bounded in $(0,\infty)$ Solution: Using L'Hospital ...
2
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1answer
18 views

If $X \geq X_t$ why is $\frac{X}{(1+|X|)} \geq \frac{X_t}{(1+|X_t|)}$? So a monotone, 1-1 transformation doesn't affect the inequality?

I am wondering why if $X=\sup_t \{ X_t \} $ for $t \in T$ which is some index set, we have that $\frac{X}{1+|X|} \geq \frac{X_t}{1+|X_t|}$. Clearly, $ \ast X \geq X_t \forall t$. My beginning is to ...
1
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0answers
16 views

Rayleigh quotient strictly increases

Consider the Rayleigh quotient $$\lambda_{L} := \max_{u \in H^{1}_{0}([0, L])}\frac{-\int_{0}^{L}u'^{2}\, dx}{\int_{0}^{L}u^{2}\, dx}.$$ Is $\lambda_{L}$ strictly increasing in $L$? Fix an $L_{1}, ...
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2answers
19 views

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
19 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 ...
0
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1answer
22 views

Does measure imply the existence of limit function?

I am reading a book and I don't quite understand some of the statements. 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
11 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
16 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 ...
0
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1answer
20 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
65 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
23 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
32 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
51 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
votes
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
69 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
34 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
votes
1answer
57 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 ...
0
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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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2answers
36 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 ...
0
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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= ...
0
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1answer
35 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 ...
1
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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
29 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
48 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
41 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 ...
1
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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
23 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
33 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 ...
6
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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
votes
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
votes
1answer
59 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
votes
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
votes
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 ...