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, integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
0answers
6 views

proof for equality of measures for case the case that $F_{\mu}$ be the completion of $F$

Let $(\Omega,F, \mu)$ be a measure space, and let $F_{\mu}$ be the completion of $F$ relative to $\mu$. If $A \subset \Omega$, define: $\mu_{0}(A)=\sup \{\mu(B: B \in F, B\subset A \}$, ...
1
vote
2answers
27 views

Unbounded sequence that does not diverge to $+ \infty$ or to $- \infty$

I'm trying to find an example of sequences such that $$a_n \to + \infty \quad \text{and} \quad b_n \to 0$$ $$a_n b_n \text{ is unbounded but does not diverge to }+ \infty \text{ or} - \infty$$ Is ...
0
votes
1answer
10 views

monotone class theorem failure for a class of subsets that is not a field

show the monotone class theorem fails if $F_{0}$ is not assumed to be a field. Monotone class theorem: Let $F_{0}$ be a field of subsets of $\Omega$, and $C$ a class of subsets of $\Omega$ that is ...
1
vote
0answers
22 views

A density question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \mod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?
1
vote
0answers
25 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
0
votes
0answers
8 views

Unconstrained Optimal Control - $J = \frac{1}{2}x^2(2) + \frac{1}{2} \int_{0}^{2}(u^2 - 2xu)dt$

I've been given the following unconstrained optimal control problem, but I feel like I've made a mistake at some point. The system $\dot x = -x + u$, where u = u(t) is not subject to any ...
0
votes
0answers
13 views

Let $a_n = \sqrt[n]{n}$ and $b_n = a_{n+1}/a_n$. Prove that $\{b_n\}_{n=5}^\infty$ is increasing and find the limit.

Let $a_n = \sqrt[n]{n}$ and $b_n = a_{n+1}/a_n$. Prove that $\{b_n\}_{n=5}^\infty$ is increasing and find the limit. I've tried to show that this sequence is increasing without luck. ...
1
vote
2answers
27 views

A set has no accumulation point is closed

I was wondering if a set $A$ has no accumulation point, is this set $A$ closed? I think this is true but not quite sured. Here is my thought: By closed set definition: A set $A$ is closed if ...
2
votes
1answer
15 views

Show the function is uniformly continouos on (0,1).

Prove that each of the following functions is uniformly continuous on (0,1). (You may use l'Hopital's Rule). b)f(x) = xcos(1/x^2) attemp in proof: I need to use the following theorem. Theorem 3.40: ...
-2
votes
3answers
50 views

How do I prove this? (real analysis)

Prove that the half open interval (0,1/n] subset R is not compact?
0
votes
1answer
14 views

Show that: $\mid\det(A)\mid\mu^*(M)=\mu^*(A(M)) $.

Let $n\in\mathbb{N},M\subset\mathbb{R}^n$ with $\mu^*(M)$ being finite and $A$ a linear mapping, which is diagonal to the standard basis of $\mathbb{R}^n$. Show that: ...
2
votes
2answers
33 views

Show that $\int_{x=a}^{x=b} f'(x) g(x) dx=f(b)g(b)-f(a)g(a)-\int_{x=a}^{x=b} g'(x)f(x)\, dx$

I have to prove the following: Suppose $f$ and $g$ are differentiable on $[a,b]$ and $f'$ and $g'$ are integrable on $[a,b]$. Prove that $f'g$ and $g'f$ are integrable on $[a,b]$ and that of: $$ ...
1
vote
1answer
19 views

Measurable functional calculus

I am struggeling with this exercise: Let $T \in L(H)$ be a self-adjoint operator and $\Psi$ be a measurable (Borel) functional calculus on the spectrum of $T$. For a Borel set $\Delta \subset \sigma ...
0
votes
0answers
13 views

Equality of inner measure and outer measure for complete measure space

Let $(\Omega,F, \mu)$ be a measure space, and let $F_{\mu}$ be the completion of $F$ relative to $\mu$. If $A \subset \Omega$, define: $\mu_{0}(A)=\sup \{\mu(B: B \in F, B\subset A \}$, ...
0
votes
3answers
45 views

Prove that the sequence $a_{n+1}=\sqrt{2a_n+3},$ $a_1=1$, is bounded.

Prove that the sequence $a_{n+1}=\sqrt{2a_n+3},$ $a_1=1$, is bounded. Proof: it's increasing and bounded above by $2$. Is that right?
5
votes
0answers
28 views

Constructing a Borel set A on R such that $0<m(A \cap I) < m(I)$ for all intervals $I$. [duplicate]

I need help constructing a Borel set $A$ on $\mathbb{R}$ with the following property: For every open interval $I$, $$0<m(A \cap I)< m(I)$$ A obviously needs to be dense in $\mathbb{R}$ and it ...
0
votes
1answer
35 views

Simplify $|2x^{2}+3x-2| $ so we can obtain it and control it in terms of $|x-1|$

First example I worked, I had $|2x^2 + x - 3|$ after some manipulations and simplifications I obtain: $|x-1|(2|x-1|+5)$. The final answer is in terms of $|x-1|$ with multiplication between two ...
0
votes
1answer
18 views

If a function is radial, then its Hardy-Littlewood maximal function is radial as well

I'm looking for a proof of the following statement: $$f\in L(\mathbb R^n)\ \text{ radial } \implies f^* \ \text{ radial}$$ where $f^*$ is the Hardy-Littlewood maximal function defined by: $$f^*(x)= ...
0
votes
1answer
15 views

Describe smallest algebra, monotone class, $\sigma$-algebra

I'm trying to understand better the concepts of monotone classes, algebras and $\sigma$-algebras so I came into the following problem. For the family $E := \{∅, \mathbb{N}, \{2\}, \{2, 4\}, \{2, 4, ...
1
vote
0answers
17 views

If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$.

If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$. Is it possible to prove this theorem?
6
votes
2answers
76 views

Is every open set a continuous image of a closed set? (in Euclidean space)

Let $A \subset \mathbb{R}^n$ be an open subset. The question is whether $A$ can always be written as a continuous image of a closed subset of euclidean space $f(C) = A$ for some closed ...
0
votes
1answer
43 views

Which of the ordered field axioms fail for the irrational numbers?

I think that the following fail: There exists an irrational number denoted $0$ such that $a+0=a$ for all irrational numbers $a$, because there is no such irrational number. For all irrational ...
1
vote
1answer
28 views

A function which is not identically zero has positive integral for some ball.

Assume that $f$ is integrable on $\mathbb{R}^d$, and $f$ is not identically zero. The hint in my book is telling me that there exists some ball such that $\int_{B} |f| > 0$ Suppose $f= ...
1
vote
1answer
22 views

Lp spaces are nested but then why is 1/x square summable but not summable?

If $1\leq s<r<\infty$ and $f\in L^r$ then $f\in L^s$, so then why is $\frac{1}{x}$ not in $L^1$ but is in $L^2$ for the counting measure $c:\mathbb{N}\rightarrow \mathbb{R}$?
1
vote
1answer
18 views

If $u$ and $\partial_n u$ along a curve are known, then the full gradient $\nabla u$ is also known

I just stumbled above this statement. If a curve $\Gamma$ in $\mathbb{R}^2$ is given. Then if $u$ and $\partial_n u$ on $\Gamma$ are known. Then the full gradient $\nabla u$ is also known. So far, I ...
1
vote
1answer
28 views

Hardy–Littlewood maximal function – an example

Let $$f(x)= \begin{cases} \frac{1}{x\log^2x}, & \text{if} \hspace{2mm} 0 < x < \frac{1}{2}\\ 0, & \text{otherwise} \end{cases} $$ I have so far shown that $f$ is integrable. However, I ...
0
votes
1answer
17 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where ...
0
votes
2answers
35 views

tricky telescopic sum

Consider the sum of $\frac{1}{n+a-b}-\frac{1}{n+a}$ from $n=1$ to infinity. Show that is in fact an finite sum. I have written down some terms but can't see where cancellation is occurring.
0
votes
2answers
22 views

Product of bounded and convergent to $0$ sequence is a convergent to $0$ sequence

Let $a_n$ be a null sequence and let $b_n$ be a bounded sequence. Prove that $a_n \cdot b_n$ is a null sequence. I tried using the product rule of sequences but cannot because $b_n$ is not ...
0
votes
0answers
14 views

Exponential Averaging Asymptotic Inequality

Let $\lambda_1(t)$ and $\lambda_2(t)$ be nonnegative integrable functions on $[0,\infty)$. Consider the averaging function of $\lambda_1$ $$k(t) = \frac{\int_0^t \lambda_1 e^{-\int_0^s ...
0
votes
2answers
15 views

Converges of a sequences defined through a continued fraction

Consider the following sequence $(b_n)_{n \geq 1}$ recursively given through the continued fraction $b_1 = \frac{1}{1}$, $b_2 = \frac{1}{1+ \frac{1}{2}}$, $ \dots , b_n = \frac{1}{1+b_{n-1}}$ ...
2
votes
0answers
26 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
1
vote
1answer
25 views

Sequence, subsequences, sub-subsequences in metric spaces

Let $(X,d)$ be a metric space, and let $\{x_n\}_{n \in \mathbb{N}}$ be a sequence in $X$. Assume that every subsequence of $\{x_n\}_{n \in \mathbb{N}}$ has a sub-subsequence that converges to the same ...
0
votes
2answers
36 views

Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
0
votes
2answers
38 views

Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy.

Let $f:[0,2]\to\mathbb{R}$ be a regulated function. Let $(x_n)_{n\geqslant1}$ be a sequence in $[0,1)$ with $\lim_{n\to \infty}x_n=1$. Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy. I ...
0
votes
3answers
66 views

How to evaluate this $1/n$ infinite sum?

How to evaluate$$\sum ^{\infty}_{n=1} {e}^{-n}$$ without using the easy-formula. We easily notice a pattern. $$\begin{align} S_1 &= e^{-1} \\ S_2 &= e^{-2} + e^{-1} = \frac{1 + e}{e^2} \\ ...
1
vote
3answers
41 views

Limit of a function with rational numbers

$$ f(x)= \begin{cases} 1-x & if x\in\mathbb{Q}\\ x & if x\not\in\mathbb{Q} \end{cases} $$ Determine $\lim_{x\rightarrow 0}f(x)$ if it exists. How do I approach this question? It seems obvious ...
0
votes
2answers
24 views

terms asymptotically equal?

I need to prove that $\left ( \frac{k-i+1}{k}\right )^{j}$ and $\left ( \frac{k-i}{k}\right )^{j}$ are asymptotically equal when k is large enough, $1\le i\le Q$ and $Q$ is a constant. Could you ...
1
vote
0answers
17 views

Lebesgue outer measure of linear subspace

Prove that $\lambda^{*}(A)=0$, where $\lambda^{*}$ is $n$-dimensional Lebesgue measure on $\mathbb{R}^n$ and $A$ is $k$-dimensional subspace of $\mathbb{R}^n$ and $k<n$. I've proved this for ...
0
votes
0answers
13 views

Proof that every subset of extended real numbers has in set of extended real numbers supremum and infinum

I need to prove this statement, but I don't know how to prove it formally correctly. Can someone help how to prove it formally? We have linear ordering $(\mathbb R^{*},<)$, where $\mathbb R^{*} ...
1
vote
2answers
29 views

Is the limit finite?

I need to find $r>0$ for which the following limit is finite $$\lim_{n \rightarrow \infty} \frac{n^{r-1}}{n^r+k^r}$$ I get inconclusiveness using the ratio test. The root test does not seem to ...
0
votes
1answer
10 views

Continuous function and constant sign

Let $f:(a,b)\to\mathbb{R}$ be a continuous function, and $t_0\in (a,b)$ such that $f(t_0)=0$. Is it true that one can find in any case some $\epsilon >0$ such that $f$ has constant sign on ...
1
vote
0answers
18 views

Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
1
vote
2answers
30 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ [on hold]

Is the following True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$ a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. EDIT: I found ...
1
vote
1answer
24 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
0
votes
1answer
29 views

A simple question in calculus (equivalence of limits).

So I want to prove the next equivalence: where D-lim, is $\lim_{n\rightarrow \infty , \ n \notin M \subset \mathbb{N}}$. The easy part, mainly $\Rightarrow$ I did I think good. I am having ...
3
votes
0answers
49 views

Prove this inequality.

Let n $\in\mathbb{N} $ and $x_0,x_1,.....,x_n $ $x_i\in\mathbb{R}$ ,$x_i>0 $ so that $ x_0 + x_1 + .... x_n =1$. Prove that for all $a\in\mathbb{R}$ and $a>0$ the inequality is verified : $\ ...
1
vote
1answer
38 views

Prove that f(x) is regulated.

Define $f:[0,1]\to \mathbb{R}$, $f(x):=0$ if $x\notin \mathbb{Q}$, $f(p/q):=1/q$, $q>0$, $p, q$ coprime integers. Prove that $f$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated ...
0
votes
1answer
29 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
-1
votes
1answer
39 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $? [on hold]

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?