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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4
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24 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 ...
-1
votes
3answers
25 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
26 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
vote
1answer
14 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 ...
0
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0answers
11 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 $\mathbb{R}^n$.
1
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3answers
24 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 ...
0
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2answers
46 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 ...
0
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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 ...
1
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1answer
22 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) ...
0
votes
0answers
16 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 ...
1
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0answers
16 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? ...
0
votes
1answer
31 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
votes
5answers
67 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
vote
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
23 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 ...
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0answers
25 views

On the integration of a Lebague measurable function [on hold]

Sincerely need help on this question, anyone who has any ideas please don't hesitate to tell me :)
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 ...
4
votes
3answers
76 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
votes
3answers
94 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 ...
3
votes
2answers
40 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
votes
3answers
49 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
votes
1answer
38 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 ...
-3
votes
1answer
31 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...
0
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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, ...
1
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1answer
61 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
votes
1answer
19 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
votes
1answer
28 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
vote
1answer
23 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: ...
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2answers
57 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
votes
3answers
101 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
votes
1answer
30 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 ...
3
votes
1answer
20 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
1
vote
1answer
30 views

Divergent or convergent but how ??

I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...
5
votes
1answer
36 views

Show that the image of Lipschitz function $\gamma : [0,1] \to R^n$ has measure $0$, if $n \ge 2$.

Problem Statement: Let $\Gamma$ be the image of a Lipschitz continuous function $\gamma : [0,1] \to R^n$, that is, $\Gamma = \{\gamma(t) : t \in [0,1]\}$, and $|\gamma(t_1) - \gamma(t_2)| \le K |t_1 - ...
1
vote
0answers
20 views

Bound on the mean value of function involving Hilbert transform

Consider the integral $$\int_{-\infty}^{\infty} x|A|^2_x\mathbb{H}(|A|^2_x) \ dx,$$ where $A=A(x,t)$ is a complex valued, compact function (I mean this in the heuristic sense that $A$ vanishes ...
3
votes
4answers
51 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then ...
1
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1answer
42 views

Difficulty in understanding converse part of proof of a propostion in Andrew Browder's Mathematical Analysis

Proposition: Let $\mu$ be finitely additive set function, defined on the algebra $\mathscr A$. Then $\mu$ is countably additive if and only if its has following property: if $A_n \in \mathscr A$ ...
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2answers
39 views

Question on Taylor series in real analysis

Suppose that $$ f(x) = \begin{cases} e^{-1/x^2} & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases} $$ How do I prove that $(d/dx)f$ at $0$?? I tried it this way, \begin{align} f'(0) & = ...
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0answers
19 views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
0
votes
0answers
22 views

Is the function ${e^{-{1}\over{x}}}\over {x}$ on $(0,1)$ uniformly continuous or bounded?

$$f(x)= {{e^{-{1}\over{x}}}\over {x}}$$ for $x\in (0,1)$ . Is this function $a$) uniformly continuous $b$) bounded but not continuous $c$) unbounded This would be uniformly ...
1
vote
1answer
29 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
0
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1answer
45 views

Give the example of compact set with infinite countable derived set [on hold]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
0
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1answer
34 views

An inequality involving the AM-GM inequality: $| x + \frac1x | \ge 2 $ (for $x<0$).

Suppose $x \neq 0 $, then $| x + \frac{1}{x} | \geq 2 $. I have shown this using the am gm inequality $(a+b)/2 \geq \sqrt{ab} $. In fact, with $a = x^2 $ and $b=1$ works. So, for $x > 0 $ we have ...
0
votes
0answers
60 views

Prove √2 exists by Archimedean Axiom [duplicate]

I am trying to prove the existence of the square root of 2. The proof: Let $$S=\{x \in \mathbb{R} ∣x \ge 0, x^2 < 2\}.$$ I understand the proof of LUB, $\alpha$ and so I am at the step where ...
0
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1answer
20 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
0
votes
1answer
24 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
0
votes
0answers
20 views

Property of nth root

I'm trying to prove the following result: "Let $x,y \geq 0$ be non-negative reals, an let $n,m \geq 1$ be positive integers. If $y=x^{1/n}$ then $y^n=x$." $x^{1/n}:=sup \{y \in \mathbb{R}: y \geq 0, ...
1
vote
1answer
14 views

checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any ...
0
votes
0answers
10 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...