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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20 views

Is Lebsegue Measure Translation Invariant?

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X\subseteq\mathbb{R}$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. ...
1
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2answers
14 views

Sequence of Partial Sums is Convergent

How do I show that the series $\sum_{n=1}^{\infty} \frac1{(2n-1)^{n}} + \frac1{(2n)^{3 }} $ is convergent? I'm trying to use Comparison Test but I'm having a hard time looking for a convergent series ...
1
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1answer
26 views

Prove that a subset of a set of measure zero has measure zero

Thm Prove that a subset of a set of measure zero has measure zero. I attempted the proof, corrections appreciated. Pf Let $A=\{x_1,....,x_N\}$ be a finite set, and let $\epsilon > 0$ be given. ...
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1answer
23 views

Convergence of Series

Prove or disprove: If $\sum_{n=1}^{\infty} x_n$ is convergent, then $\sum_{n=1}^{\infty} (-1)^{n} x_n $ is convergent.
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4answers
50 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
3
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0answers
33 views

How to show a function is not riemann integrable

I am wondering how I can show the following function is not Riemann integrable. Since we are on a closed and bounded interval I couldn't use that it is unbounded, etc ( think) ...
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1answer
22 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ ...
2
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1answer
39 views

Suppose that $f_n \colon \Bbb{R}\to [0,1]$ for $n\in \Bbb{N}$ and that each one of these functions is decreasing

Suppose that $f_n \colon \Bbb{R}\to [0,1]$ for $n\in \Bbb{N}$ and that each one of these functions is nondecreasing. Prove that there exists a function $g\colon \Bbb{R} \to [0,1]$, a countable set ...
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0answers
13 views

how to show the equivalence of density

How to show $E$ is dense if and only if $int(\mathbb{R}- E) = \emptyset$ suggestions please. I do not see how to a direct proff
2
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1answer
19 views

Show that $fg$ is differentiable at $\hat{x}$ and that $(fg)'(\hat{x})= g(\hat{x})f'(\hat{x}) + f(\hat{x})g'(\hat{x})$

Let $U$ an open set in $\mathbb{R^n}$, $\hat{x} \in U$ and let $f : U \to \mathbb{R}$ and $g : U \to \mathbb{R}$ two different differentiable functions at $\hat{x}$. Show that $fg$ is ...
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0answers
5 views

Tradeoff between different formulations of partition of unity

In Wikipedia, a partition of unity of a topological space $X$ is a set $R$ of continuous functions from $X$ to the unit interval $[0,1]$ such that for every point, $x\in X$, there is a neighbourhood ...
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0answers
10 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuous diff'able function on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We ...
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0answers
17 views

Connected sets prove that definitions are equivalent

I found the following two definitions of connected set. I couldn't really see how they were equivalent so I tried to prove it. Definition: Two subsets $A$ and $B$ of a metric space $X$ are said to ...
3
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1answer
9 views

Proving weak coercivity by young's and interpolation inequalities

Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some ...
1
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1answer
15 views

Infimum of the sum of absolute differences from $x$ occurs when $x$ is the median [duplicate]

You have a set of real numbers $\{x_i\}_{i=1}^n$. You want to find what $x_{min}$ that minimizes $$\sum_{i=1}^n |x_i - x_{min}|$$ I can argue that this is minimized when $x_{min}$ is the median ...
2
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2answers
61 views

Does $\frac{\sin(x\ln x)}{x\ln x}$ as $x$ approaches $0$ from the right side have a limit?

I am trying to determine if $$\lim_{x\rightarrow0^{+}}\frac{\sin(x \cdot \ln(x))}{x\cdot \ln(x)}$$ has a limit ? Since $$\ln(x) \rightarrow -\infty\text{ as }x \rightarrow 0^{+}$$ I have tried ...
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0answers
11 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
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1answer
20 views

Proving that the sequence converges

I would like some help with the following problem. Thanks for any help in advance. Let $(x_n)$ and $(y_n)$ be convergent sequences of positive real numbers. Let $ x_n \xrightarrow[n \to \infty]{} x$ ...
3
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1answer
16 views

Which values of $p$, $f$ is it differentiable at the point $(0,0)$?

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
2
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0answers
24 views

Picard iteration for second order ODE

Any help for the following two questions will be much appreciated: Question 1: Consder the ODE with IVP $y''(t)+g(t,y)=0, \ y(0)=y_0,\ y'(0)=z_0$ where g is continues on some region $D$, ...
3
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2answers
35 views

How to obtain $\lim_{(x,y) \to (0,0)} (x^2·y^3)/(x^4+y^6)$

I want to determine $$\lim\limits_{(x,y) \to (0,0)} \frac{x^2·y^3}{x^4+y^6}$$ I'm sure the limit exists, it's zero because I tried to find other different limits in line and parabola points ($(x,mx)$ ...
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0answers
15 views

Change of variables from unit unit ball to another ball for integration

What is the general formula for changing coordinates for integration from the unit ball to another ball? For example, if I wanted to change from integrating $f(x-r)$ over $B(0,1)$, the open ball about ...
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1answer
17 views

Triangle inequality with a twist

Assume $t>0$ and $x,y,z\in [0,t)$ how would one go about showing $$\min \{|x-y|,t-|x-y|\}\leq\min \{|x-z|,t-|x-z|\}+\min \{|z-y|,t-|z-y|\} $$ If the first one materializes from every minimum, then ...
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0answers
12 views

Limit of ratio of quadratic forms

Let $Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R$ be quadratic forms. Find a necessary and sufficient condition for $\lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)}$ to exist in ...
0
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1answer
28 views

For every normed space the norm map is not Fréchet differentiable at $0$.

Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$. Not really sure where to start ...
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2answers
16 views

Prove that $C^1[0,1]$ is space of continuously differentaible function with $C_1$ norm is separable.

$C^1[0,1]$ is space of continuously differentiable function with $C_1$ norm.Then the space $ (C^1[0, 1],)$ is a separable space. I am thinking of c^1[0,1] is subset of c[0,1], and c[0,1] is separable. ...
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2answers
47 views

Show that $f$ is not differentiable at $(0,0)$ - $\frac{x_1^2x_2}{x_1^2+x_2^2}$

Let the function $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x_1^2x_2}{x_1^2+x_2^2}, & \quad \text{if } (x_1,x_2) \not= 0 \\ 0, & \quad ...
0
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0answers
24 views

Give an example to show that $f_n$ fails to converge to $f$ uniformly over $S$ if $S$ is not compact

Given the theorem: Suppose $S \subset \Bbb R^n$ is compact, and $P$ is an equicontinuous sequence of functions ($f_n$) over $S$ converging pointwise to a function $f$ at each $x \in S$, then $f_n$ ...
3
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0answers
27 views

Check the proof of $\Bbb R$ as set of subsequential limits

I want to prove that there is a sequence in $\Bbb R$ that has all of $\Bbb R$ as its set of subsequential limits. Could someone help me check my proof? If it's not correct, could someone give a proof? ...
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1answer
27 views

Stuck on Applying Cauchy Convergence Criterion - Limit Theory

I get stuck on the following problem for a rather long time that I finally decide to ask for help. The problem is as below: Determine whether the sequence converges by applying Cauchy Convergence ...
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votes
4answers
103 views

What is $\lim_{(x,y) \to (0,0)} \arctan(xy)/\sqrt{x^2+y^2}$?

The limit is this: $$\lim\limits_{(x,y) \to (0,0)} \frac{\arctan(xy)}{\sqrt{x^2+y^2}}$$ It's not necessary to give a whole solution, I want the path to see how to solve it. I tried both with ...
4
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2answers
243 views

The set of integers is not open or is open

Baby Rudin gives the example of the set of all integers being not open if it is a subset of $\mathbb{R}^2$ (I forgot how to code the symbols on this site) If we consider the set of integers in ...
1
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1answer
25 views

Pointwise or Uniform convergence?

Consider the functions $f_n:[-1,1]\to\mathbb{R}$ defined by $$f_n(x):= \frac{x}{\sqrt{x^2 + \tfrac 1n}}$$ and determine whether the convergence is uniform or pointwise. I can see that this will ...
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1answer
29 views

Is every topological space is measurable?

Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
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1answer
50 views

when $\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k$ exist?

When does $$\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k\ \ ?$$ My first way : Since $$\sum_{k=0}^n\binom{n}{k}x^k=(1+x)^n$$ the limit exist when $x\in ]-2,0]$ et it's limits is $0$ when $x\in ...
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1answer
32 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
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0answers
45 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
0
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0answers
29 views

Conditions for a function to lie in $L^p(\mathbb{R})$ [on hold]

Let $(X, \mathfrak{M})$ be a measurable space. What are some sufficient and necessary conditions for a function $f : X \to \mathbb{R}$ to lie in $L^p(\mathbb{R})$ for $p \in [1,\infty]$? Is true ...
3
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0answers
44 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
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1answer
18 views

Oscillation of a function at a point

Why do we need to take open neighbourhoods around the point in consideration while defining oscillation of a function at that point? (We're working in R) For ref. Bartle & sherbert(introduction to ...
0
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1answer
28 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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2answers
33 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
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1answer
37 views

Distinct real roots .

Problem : If $|\log(x)| - px = 0$ has three distinct real roots then the range of $p$ will be ? My attempt : I tried to see the problem graphically and made the graph. So I am able to see that ...
1
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1answer
22 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
1
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1answer
25 views

Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
0
votes
3answers
94 views

Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence [on hold]

Let $$ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} \quad (n \in \mathbb{N}). $$ Show that $a_n$ is not a Cauchy sequence even though $$ \lim_{n \to \infty} a_{n+1} - a_n = 0 ...
1
vote
1answer
39 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
2
votes
1answer
24 views

Show that $f$ is differentiable at point $x \not= (0,0)$ - $h(x) = (\sin ||x||)^p \cos \frac{1}{||x||}$

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
0
votes
1answer
25 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...