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.

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A countability problem of $\mathbb{R}$

I am having a really hard time coming up with a proof for this problem. For every finite set $F\subseteq \mathbb{R}$, let $\Sigma(F)$ denote the sum of the numbers in $F.(\Sigma(\emptyset)=0)$. Show ...
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1answer
38 views

hexic polynomial question

I am faced with a polynomial of the form $$ ax^6+bx^3+cx+d=0, $$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
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0answers
27 views

Question about the definiton of addition of Dedekind cuts

Given two Dedekind cuts alpha and beta, their addition is defined to be the set {a+b | a belongs to alpha and b belongs to beta}. But why does this define addition? Why does this deserve to be called ...
2
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1answer
64 views

What exactly is a product measure?

If we have $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ a complete measure space with underlying complete spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$, and $\lambda = \mu \times \nu$, what ...
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0answers
17 views

Is $d\mu = g\, dm$ a Radon measure for $g \in L^{1}(dm)$?

Let $m$ is the standard Lebesgue measure and let $g \in L^{1}(dm)$. We know that $m$ is a Radon measure. Is $\mu$ defined by $d\mu = g\, dm$ also a Radon measure? We first claim that $\mu(K) < ...
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1answer
40 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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1answer
19 views

Prove a set which conatins one point from each class in circle of circumference 1 is nonmeasureable

(Kolmogrov,p268,problem 7) Let C be a circle of circumference 1 and let $\alpha\in\mathbb R\setminus\mathbb Q$. Let all points f C which can be obtained from each other by rotating C through an ...
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0answers
57 views

$a_{n+2} - a_n$ converges to $0$. Prove $\frac {a_{n+1}-a_n}{n}$ converges to $0$ [closed]

If $a_{n+2} - a_n$ converges to $0$. Prove $$\lim_{n\rightarrow\infty}\frac {a_{n+1}-a_n}{n} = 0$$
2
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1answer
64 views

Inequality of intervals lengths.

Let $(I_j)_{j = 1}^n$ be a finite collection of intervals that covers the rationals in $[0, 1]$. Prove that $\sum_{j = 1}^n \ell (I_j) \geq 1$. (Here, $\ell (I)$ denotes the length of the interval ...
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1answer
31 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
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46 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
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1answer
22 views

continuous on $[0,\infty)$ and uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , to show uniform continuity on $[0, \infty)$

Let $f:[0, \infty) \to \mathbb R$ be a continuous function which is uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , then how to show that $f:[0, \infty) \to \mathbb R$ is ...
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1answer
22 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
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Why is $0^0$ undefined when $x^x=1$ as $x$ approaches $0$?

This question was born in another post available here. I believe $0^0=1$, because $x^x$ is continuous as $x$ approaches $0$. Consider $\lim_{x \to 0}x^x$. Let ...
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1answer
21 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
2
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1answer
39 views

Divergence-Convergence of the sequence $\sin(n!{\pi}\theta)$

I am working on the convergence-Divergence of $\sin(n!{\pi}\theta).$ In his book, Hardy(A Course of Pure Mathematics) page 128 cited " The case in which $\theta$ is irrational cannot be dealt with ...
2
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1answer
33 views

Proving the Weierstrass M-Test with topology

I've encountered some theorems in analysis that are ultimately provable in a more elegant way with topology. So, is there a topological proof of the Weierstrass M-Test, ideally not using terribly ...
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1answer
46 views

Continuity of a piecewise constant function

A)I can draw the graph and see that the function is continuous at x=0.3 as when you approach it from the left and right you get the same result B) not sure how to prove properly but it is not ...
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1answer
17 views

Is this theorem about “completion of metric space” correct?

It's well-known that there is a completion of a metric space unique upto isometry. I have tried to modify this theorem slightly and I proved this statement: Let $(X,d_X)$ be a metric space. ...
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1answer
23 views

Convergence of this Sequence of Functions to its Supremum [closed]

Let f be positive and continuous on $J=[a,b]$. Let M=sup{ f(x) : x $\in$ J}. Show that : $$ M=\lim_{n}\left ( \int_{a}^{b}(f(x))^ndx\right )^{\frac{1}{n}}$$
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1answer
31 views

Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
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constrained optimization and differential equation

Consider the following differential equation system (cylindrical coordinate system): $\frac{dP_x}{dz} = P_x C \int\limits_0^{2\pi}\int\limits_0^a \frac{f(r, \theta)}{g(r, \theta, z)} r dr d\theta$ ...
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1answer
48 views

Is liminf of a product equal to the product of liminfs?

My question is just for curiosity. I was thinking if is true this curious affirmation: Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then ...
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2answers
41 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
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0answers
27 views

$l^1$ and $l^{\infty}$ not isometric in general [duplicate]

I'm trying to figure out the following example: Consider the map $\Lambda:l^1\rightarrow (l^{\infty})^{\ast}$ given by $$\Lambda(\xi)(\eta_1,\eta_2,\cdots)=\sum_i \xi_i\eta_i$$ I can show that it is ...
2
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1answer
31 views

If $f \in L^{1}(d\mu)$, is it true that $\int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu)$?

Ok, so we have $f \in L^{1}(d\mu)$, with $(X, \Sigma, \mu)$ a complete measure space. If we assume $f$ is nonnegative, we can define a measure $\rho(E) = \int \limits_{E} f \,d\mu$ for $E \in ...
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68 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
3
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1answer
40 views

If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
2
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1answer
62 views

How to express $x^5$ as a telescoping series

How do I express $x^5$ as a telescoping series (i.e, $x^5=x^5-x^{a}+x^{a}-x^b+x^b-...)$? In other words, I must find functions $b(n),c(n),d(n)e(n)$ such that ...
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1answer
33 views

Monotone increasing function has finite one-sided limits at every point

"Let $f:(a,b)\rightarrow\mathbb{R}$ be a monotonous increasing function on $(a,b)$ and let $x_0\in(a,b)$. Show that $f$ has finite limits $f(x_0^-)=\lim_{x\rightarrow x_0^-}f(x)$, ...
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2answers
42 views

Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
3
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1answer
46 views

Measure Theory Inequality

I was having trouble showing the following inequality: Prove that if $A \subset I = [0,1]$ has measure $u(A) < 1$ and $\epsilon > 0$, then there is an interval $[a,b] \subset I$ such that $u(A ...
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1answer
29 views

Lebesgue measurable functions and the absolute value of them

Let $f$ is a measurable function. If $f$ is Lebesgue integrable, is the absolute value of $f$ Lebesgue integrable?
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114 views

Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
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1answer
26 views

A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
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0answers
15 views

Fourier series and irrational period

I'd appreciate it if I could get a hint to solve the following exercise. Let $\alpha$ be an irrational number and let $f$ be a real valued measurable function on $\mathbb{R}$ such that ...
3
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1answer
65 views

Analytical solution of a polynomial $a\cdot x^{e}+b\cdot x^{4\cdot e}+c =0$

Is it possible to get an analytical solution of the equation $a\cdot x^{2\cdot e}+b\cdot x^{e+1}+c =0$ Which can be also written as (due to the value of $e$): $a\cdot x^{e}+b\cdot x^{4\cdot e}+c ...
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2answers
48 views

$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
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1answer
38 views

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence?

Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that ...
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0answers
28 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
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0answers
29 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
4
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1answer
35 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
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3answers
353 views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
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1answer
29 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
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3answers
76 views

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof

For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's ...
2
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1answer
58 views

Point-wise converging convex functions on $[0,1]$

Suppose we have a sequence of continuous convex functions $\{f_n\}$ defined on $[0,1]$ which converge point-wise to a limit $f$ on $[0,1]$, i.e. for all $x \in [0,1]$ $$\lim_n f_n(x) = f(x).$$ Let $G ...
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0answers
99 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
1
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1answer
29 views

If $E, F \subset [0, 1]$, $m(E), m(F) > 0$, and $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$, show $m(F\cap E_n) > 0$ for sufficiently large $n$

Suppose $E \subset [0, 1]$ has positive Lebesgue measure and let $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$. If $F \subset [0, 1]$ has positive Lebesgue measure, show that so does $F \cap E_n$ for ...
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1answer
39 views

Inequality involving functions bounded above but not below: $|\sup_{x\in\mathbb R}f(x)-\sup_{x\in\mathbb R}g(x)|\leq\sup_{x\in\mathbb R}|f(x)-g(x)|$?

Let $ f,g: \mathbb{R} \rightarrow \mathbb{R} $ be bounded above and below, then we can prove that: $$ | \sup_{x \in \mathbb{R} } f(x) - \sup_{x \in \mathbb{R} } g(x) | \leq \sup_{x\in \mathbb{R}} | ...
1
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1answer
55 views

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates $$ \left\{ \begin{array}{c} u = yz\sin(x)\\ v = y^2 - x\\ w = xz \end{array} \right. $$ Determine the ranks ...