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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Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
2
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1answer
53 views

How do I show $(a, b)$ is incomplete?

If I take the sequence $\{a + 1/n\}$, then it's Cauchy and the limit as $n$ goes to infinity is $a$, which completes the proof. Is this correct? Edit: I doubt this is the right approach because the ...
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3answers
26 views

Open intervals in $R^1$ is open

I know this question would seem like a duplicate, but I here I provided a proof of the statement I just don't know how to justify certain thing in my proof. Proof: Suppose y is an arbitrarily ...
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1answer
25 views

Uniform convergency of a two sequence of functions

For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$ Then, which is(/are) correct ? (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ ...
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0answers
10 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
2
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3answers
49 views

$0,1,0,1,0,1$… has only $2$ limit points

Prove that the sequence $0,1,0,1,0,1$... has only $2$ limit points : $0$ and $1$. To be frank, I know the solution to the above particular problem. What I am interested is in knowing a general ...
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2answers
76 views

Under what conditions is $|x+y|=|x|+|y|$ true?

What instance that this equation would be true? $|x+y|=|x|+|y|$ Given that $x$, $y$ are elements of real numbers.
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2answers
26 views

Finding error in reasoning about geometric series sum

A supposedly fast method to find the sum of a geometric series is the following one. Let $$S = \sum_{n = 0}^{+\infty} q^n$$ then $$S = 1 + q\left[\sum_{n = 0}^{+\infty} q^n\right] = 1 + qS.$$ Hence ...
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0answers
14 views

Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
4
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1answer
48 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
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1answer
34 views

Help me solve this problem. [on hold]

Under what condition is it true that $|x-y|+|y-z|=|x-z|$ ?
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40 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
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2answers
71 views

Real Analysis: if the integral of the cube of a function exists, does it follow that the integral of the function also exists?

Let $I=[a,b]$. Given that $\int_a^bf^3(x)\;dx$ exists, does it follow that $\int_a^bf(x)\;dx$ exists? Let's let $a$ and $b$ be real numbers.
4
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1answer
110 views

Find all values such that the series converge:

Find all the values of $p\in\mathbb R$ such that the following series converge: $$\sum_{k=2}^\infty (\log k)^{p\log k}$$ I would like hints only. I've tried using the exponential function ...
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1answer
38 views

Prove this sequence is bounded

This is not any exercise on itself, but I was reading a proof in which a sequence similar to this appeared: $$a_n=n\lambda^{n-1}, \quad|\lambda|<1$$ In essence. Then I came across the assertion, ...
2
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2answers
36 views

Show That $\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$

My question concerns proving an inequality between two extreme limits, namely: $$\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$$ Using the following defintions: Let $f: E \to \mathbb{R}$ be a ...
2
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1answer
20 views

Proving that $f \big|_{\partial A} = 0$, where $A = [f> 0]$.

I found this exercise: let $M$ be a metric space, and $f: M \to \Bbb R$ be a function, and $A = \{ x \in M \mid f(x) > 0\}$. Prove that if $x \in \partial A$, then $f(x) = 0$. I think that we must ...
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0answers
17 views

A question regarding separable continuity and measurability

Suppose $f(x,y)$ is a function mapping from $R^2$ to $R$ and it is continuous in each variable separately (separable continuity), then why $f(\frac{{\left\lfloor {mx} \right\rfloor ...
14
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2answers
99 views

$\|f(x)-f(y)\|\geq ||x-y||$

$K$ is a compact subset of $\Bbb R^n$ and $f:K\rightarrow K $ satisfies : $$\|f(x)-f(y)\|\geq \|x-y\|$$ Show that $f$ is bijective, and that : $$\|f(x)-f(y)\| = \|x-y\| $$ It's easy to show that ...
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2answers
18 views

Deriving convexity from Taylor series expansion

Why is the function $f(x) = \sum^\infty_{k=1} (3x)^{2k}$ convex? What is the condition on the coefficients to deduce that $f$ convex?
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0answers
17 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
2
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1answer
37 views

literature on advanced calculus [on hold]

I need your opinions on this particular textbook: Advanced Calculus by Robert C. Buck. In my first year in college I finished two semesters of single-variable calculus and now I'm looking for a proper ...
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4answers
28 views

If $d(x,y)$ is a metric, how does the following inequality apply?

I'm interested if someone can formally type out why this is. I thought it was trivial, but the professor wanted a more detailed explanation: $${d(x,y)\over {1+d(x,y)}}\leq ...
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0answers
17 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
3
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0answers
60 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma ...
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1answer
26 views

Distance between two points using Lagrange [on hold]

Set up the system of equations required according to Lagrange in order to minimize the square of the distance between P1 and P2. $${M_{1}=‎\lbrace(x,y)\in R^2} \mid x^2+\frac{9}{4}(y-1)^2=9‎\rbrace$$ ...
0
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0answers
26 views

Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })^4$.

Let $f:\Bbb{R}\to \Bbb{C}$ be an infinitely differentiable function with period T. Also, $f^{(4)}=\lambda f$. Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })$. Attempt: I did ...
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2answers
38 views

Limits and definition integrals involving logarithms

Let $a \in (0,1)$ and define $$I_n(a)=\int_a^1 (\ln x)^n \, \mathrm{d}x$$ Show that limit as $a\to 0$ we have, $$\lim_{a\to 0}I_n(a)=(-1)^n \cdot n!$$
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26 views

Evaluating MacLaurin polynomial of composite function

I want to evaluate the MacLaurin polynomial (Taylor polynomial around 0) $p$ of $f(x) = \sin(x^3)$ of order $11$ at $x=1$ and do this as efficient as possible (without much computation). When I just ...
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1answer
24 views

For $n \geq 2$, find $\theta_n, \theta_n > 1$ s.t. $-log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$

For $n \geq 2$, show that $\exists$ a number $\theta_n, \theta_n > 1$ such that $-log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$ lim$_{n\to \infty} \theta_n$ My attempt: I am not ...
2
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3answers
56 views

“less than $+\infty$” is not bounded above?

This is an exercise in my textbook. I am puzzled that since $f$ is less than $+\infty$ on $E$, doesn't it already means $f$ is bounded above? Why we still need to show it?
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1answer
27 views

proof of the singular-values of orthogonal matrix

What is a simple and intuitive proof that the singular-values of orthogonal matrix $A$ is $1$?
3
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1answer
59 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
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1answer
19 views

Integrable convex function vanishes at infinity

Why does a function that is Riemann-integrable in $[0, \infty)$ and that is convex vanishes at infinity?
2
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1answer
21 views

Is a monotone function defined on any kind of interval measurable?

Definition of Measurable set: A set $E$ measurable if $$m^*(A) = m^*(A ∩ E) + m^*(A ∪ E^c)$$ for every subset of $A$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function $f: D ...
2
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3answers
34 views

Can anyone prove the second property of a the following metric? $d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$ [on hold]

$$d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$$ $2.) d(f,g)=0 \iff f \equiv g$ Now in my notebook some lemma is called upon, concerning integrals, but it is unclearly ...
4
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1answer
47 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
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13 views

Is there a Gronwall-type lower bound inequality?

There are various versions of Gronwall's lemma. One of them is something like the following: If $f(t) \leq h(t) + \int_0^t g(s)f(s)ds$, plus some continuity conditions, then $f(t)\leq$ something ...
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If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics.

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics defined. Is the diameter of a subset unique with respect to the two most ...
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15 views

Homeomorphism Proof

The Question: Show that the map, h, defined by $h(x) = {y_{0}, y_{1}, y_{2}...,}$ which $y_{i}= \frac{x_{i}}{2}$ is a homeomorphism. ($h: K \mapsto \Sigma_{2}^{+} for x= 0.x_{0}x_{1}x_{2}...$ with ...
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0answers
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If $\partial E$ has Jordan outer measure zero, then $E$ is measurable.

I am going through Tao's measure theory book, and have to prove If $\partial E$ has Jordan outer measure zero, then $E$ is measurable. where $\partial E$ denotes the boundary of the set $E$. I ...
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0answers
29 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
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1answer
31 views

A new metric formed from the old

Let $(X,\rho)$ be a metric space. Is it true that $d:=\dfrac{\rho}{1+\rho}$ is also metric. If it's true can anybody give a hint how to prove the triangle inequality for $d$?
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6 views

About a property of the upper triangular projection of a matrix

I need a hand checking that a property about the upper triangle projection of an infinite matrix holds. $\bullet$ Let A be an infinite matrix $A=(a_{ij})_{i\geq 1\;j\geq 1}$. We define its upper ...
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5answers
85 views

Proving that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$

I was reading sequences in Terence Tao Analysis book and I came across the question: Prove that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$ In the hint it says that ... you may ...
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1answer
20 views

Non-Cauchy products for series?

Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the ...
2
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0answers
22 views

Finding a Lyapunov Function for a system involving a trigonometric function

I'm dealing with determining if $(0,0)$ is stable or not for the following system via constructing a Lyapunov function. The system is $$ \begin{cases} x'(t)=(1-x)y+x^2\sin{(x)}& \\ ...
3
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2answers
64 views

Prove for any set $E\subset R$ with lebesgue measure 1 there exists a subset with lebesgue measure 1/2.

Prove for any set $E\subset R$ with Lebesgue measure 1 there exists a subset with Lebesgue measure 1/2. It looks easy but I have tried for an hour and could not find a way to prove it. Can anyone ...
2
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1answer
45 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
2
votes
1answer
35 views

Interchanging sum and differentiation, almost everywhere

Let $\{F_i\}$ be a sequence of nonnegative increasing real functions on $[a,b]$ with $a<b$ such that $F(x):=\displaystyle\sum_{i=1}^\infty F_i(x)<\infty$ for all $x\in [a,b],$ then show ...