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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If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
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16 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
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31 views

For a finite set in $\mathbb{R}$, the interior is empty and the closure and boundary are the set itself

How do I show explicitly that for a finite set in $\mathbb{R}$ the interior is empty and the closure and boundary are the set itself? For closure is simple: it is union of boundary and interior.But ...
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1answer
15 views

Consistency of differentiable functions on a closed subset.

I have a question about differentiable functions. Let $U \neq \emptyset$ be a open subset of $\mathbb{R}^{n}$ and $F$ be a closure of $U$. I want to define the space of infinitely differentiable on ...
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1answer
31 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
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2answers
37 views

Does there exist a subsequence whose intersection has measure greater than $1/2$?

I ran across the following problem on this review guide. It is problem 1.25, though I've changed the wording slightly. The measure is implicitly Lebesgue measure. Let $E_n$ be a sequence of ...
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1answer
30 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
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24 views

Question 10 from N.L Carothers' “Real Analysis”, Chapter 11.

Let $(x_{i})$ be a sequence of numbers in $(0,1)$ such that the $\lim_{n\to \infty}(1/n)\sum_{i=1}^{n}x_{i}^k$ exists for every k=0,1,2,.... Show that $\lim_{n\to \infty}(1/n)\sum_{i=1}^{n}f(x_{i})$ ...
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1answer
20 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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1answer
26 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$.show that $\int_1^3 f = \phi(3) - \phi (1)$

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$. Then to show that $f$ is integrable and evaluating the value of $\int_1^3 f$. I have done upto this. But ...
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35 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
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21 views

Support of Radon measures

I am reading Folland's Real Analysis. The following is the exercise 7.2.b. Let $X$ be a locally compact Hausdorff space with a Radon measure $\mu$. Show $x\in\text{supp}(\mu)$ iff $\int f \text ...
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35 views

Convergence conditions

While studying for my exam in real analysis, I came up with the following problem. Given two sequences $\{a_n\}\to 0$ and $\{b_n\}\to\infty$. What should be the weakest condition so that $$\sum a_n ...
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62 views

$ \lim_{x \rightarrow \infty} e^{1/x} = a $ is not equivalent with $ \lim_{x \rightarrow \infty} a^x = e$?

I have problems understanding, why $$ \lim_{x \rightarrow \infty} e^{1/x} = a $$ is not equivalent with $$ \lim_{x \rightarrow \infty} a^x = e. $$ In the first case there is a solution $a=1$, and ...
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35 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
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2answers
25 views

Minimum distance between closed sets.

I understand that if given a compact set K and a closed set C,that are disjoint, of a metric space then it follows that there is a minimum distance between them(You can prove this via a continuous ...
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2answers
48 views

Continuity in $\mathbb R$ results in continuity in $\mathbb R^2$; Proof?

During studying of proof of some other theorem, I faced with the claim (without proof): since $f(x,t)$ and $g(x,t)$ are continuous functions [$f,g:\mathbb R^2 \rightarrow \mathbb R$] thus the ...
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1answer
19 views

Question about proof extending measure to complete measure

I am looking through a proof in Folland, for Theorem 1.9, which states: Suppose that $(X, M, \mu)$ is a measure space. Let $N = \{N' \in M : \mu(N') = 0\}$ and $M' = \{E \cup F : E \in M' \text{ and ...
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20 views

Proving inner measure equal to outer measure if a set is measurable

I'm doing the problem 19 in Real Analysis like below: Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of $E$ to be ...
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2answers
28 views

Limit of Uniformly convegent sequence of real valued functions:

I was attempting this qualifying problem. $\{f_n\}_{n=1}^{\infty}$ is a sequence of real valued functions on $\mathbb{R}$. If $ f_n $ converges to $f$ uniformly then $f$ must be continuous. I am ...
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82 views

Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous?

This is a question that came up as a true false question in my textbook, and I was wondering what you thought of my reasoning. I claim that even though a graph of such a function doesn't look ...
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1answer
48 views

let $\phi (x) =\lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is not differentiable at $1$.

For $x \geq 0$, let $\phi (x) = \lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is continuous at $1$ but not differentiable at $1$. First we calculate $\phi (x) ...
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1answer
66 views

How do I prove that $ax+by+cz=d$ has infinitely many solutions on $S^2$?

Let $a,b,c,d$ be reals such that $a^2+b^2+c^2>d^2$. How do I prove that $\{(x,y,z)\in S^2: ax+by+cz=d\}$ is infinite? This is geometrically trivial, but I'm stuck at proving it rigorously..
3
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1answer
32 views

Construct a special measurable functions

Suppose $A$ is a measurable subset of $[0,1]$, show that there exists a measurable function $f(x)$ on the interval $[0,m(A)/2]$ such that $$m([x,f(x)]\cap A)=\frac{1}{2}m(A)$$ for all $x$. I know ...
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22 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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26 views

limit of a complex expression

Suppose that $$R(r)=\left[(a_0-r^2a_2)e^{ir\tau}+(b_0-r^2b_2)\right]^{\frac{1}{r}}\;,$$ where $a_0,a_2,b_0,b_2\in\mathbb{R}$ and $\tau>0$. What is $\displaystyle\lim_{r\rightarrow\infty}R(r)$? I ...
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93 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
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1answer
19 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
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59 views

Show that the cardinality of the preimage is measurable

Suppose $f$ is continuous in $\mathbb R$ such that $|\{x:f(x)=y\}|<\infty$ for any $y\in \mathbb R$(where $|\cdots|$ means cardinality). Show that $$ g(y)=|\{x:f(x)=y\}| $$ is a measurable ...
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43 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...
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1answer
46 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
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1answer
21 views

$L^1 ([0,1])$, bouned linear functional, absolute continuous function

I am studying for an Analysis prelim and was wondering if someone could perhaps either validate or invalidate my proof for the following problem: "Let $L^1 ([0,1])$ be the space of Lebesgue ...
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0answers
36 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
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4answers
46 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
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30 views

Induction on derivatives

I have troubles understanding this induction proof: Let $$g(x) = \vert x \vert^{2k+1}$$ Show by induction: $$\frac{\partial ^N g(x)}{\partial x_{i_1} \dots \partial x_{i_N})} = cx_1n \dots x_iN \vert ...
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1answer
57 views

Condition for Continuity (two variable)

I came across the following question while studying for quals. This one is from a previous qualifier. I have a few ideas (which I'll mention below), but am stuck on how to complete the problem. Any ...
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2answers
44 views

If a measure $\mu$ is less than a measure $\nu$ on a generating $\pi$-system, can we conclude that $\mu \leq \nu$?

Let $\mu$, $\nu$ be finite measures on the non-degenerate compact interval $[a, b] \subseteq \mathbb{R}$ provided with the Borel $\sigma$-algebra. It is well-known that if $\mu(B) = \nu(B)$ for every ...
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2answers
32 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
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1answer
66 views

How to prove that $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $?

If $n \in \mathbb N$ and $n \geq 2$, then we have $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $. My try : Once if we can prove that for all $k \in ...
3
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53 views

Is this statement “a map $f$ is continuous if and only if for any open set $G$, ${f^{ - 1}}(G)$ is still open” true?

I am really puzzled by this statement and it has so many different versions in different places. Yesterday I did a homework to prove that a finite function $f$ is continuous if and only if ${f^{ - ...
3
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3answers
65 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
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1answer
14 views

Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
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1answer
12 views

Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
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1answer
27 views

Proving Squeeze Theorem using Order Limit

Would this be a valid way of proving the squeeze theorem using the Order Limit theorem? If $x_n \leq y_n \leq z_n$ for all $n \in \mathbb{N}$ and if $\lim x_n =\lim z_n = l$, then $\lim y_n = l$ as ...
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1answer
101 views

Is $(\frac 1{n^2 \sin n })$ convergent ? If so , what is the limit? [duplicate]

Is the sequence $\left(\dfrac 1{n^2 \sin n }\right)$ convergent ? If it does, with what limit?
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35 views

Principle of infinite descent

I'm trying to prove the following: "It is not possible to have a sequence of natural numbers which is in infinite descent. (Hint: assume for sake of contradiction that you can find a sequence of ...
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20 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
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1answer
22 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
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1answer
36 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...