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

0
votes
0answers
15 views

Lemma. $\exists N\in\mathbb{N}\forall n>N: x_n=0\Leftrightarrow x=\frac{q}{p^N}$ for some $q\in\mathbb{N}.$

https://proofwiki.org/wiki/User:J_D_Bowen/Math710_HW1 in the lemma of exercise 5 in direction $\Leftarrow$: Why $S_{N-1}=x$? I dont see! Thanks!
2
votes
2answers
54 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
1answer
35 views

Computing radius of convergence for series without coefficients explicitly defined

Consider the series $$ 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} a_n (x-3)^n. $$ I'm trying to find the radius of convergence for this series. However, I'm having trouble coming up with a ...
1
vote
5answers
96 views

Existence of limit for a given sequence: $x_{n+1} \le x_n + 1/{n^2}$

Let $x_n$ be a sequence in $\mathbb{R}$ such that $$x_{n+1} \le x_n + \frac{1}{n^2}$$ Prove that $\lim x_n$ exists (it can be a real number or infinite). I've tried to prove it using the ...
1
vote
0answers
24 views

$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
1
vote
2answers
40 views

Existence of a metric space M with no continuous map from M to any other metric space

Is it possible to have a metric space M such that there is no continuous map from M to any other metric space?
0
votes
0answers
24 views

Minkowski inequality for $0<p<1$

I'm trying to prove this, $$\left ( \sum_{i=1}^{n}(x_i+y_i)^p \right) \geq \left ( \sum_{i=1}^{n}(x_i)^p \right)^\frac{1}{p} + \left ( \sum_{i=1}^{n}(y_i)^p \right)^\frac{1}{p} $$ for $0<p<1$. ...
1
vote
2answers
102 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
0
votes
1answer
87 views

If $\lim_{x \to \infty}f(x)=a$ then $\lim_{x \to \infty}f'(x)=0$ - whats wrong with the proof?

Here's how i would prove this. Since we have that $\lim_{x \to \infty}f(x)=a$ this implies that $\lim_{x \to \infty}f(x + 1) - f(x)=0$ By mean value theorem we have that $\lim_{x \to ...
12
votes
2answers
351 views
+100

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
0
votes
1answer
96 views

Few Questions about analysis in Rudins book

I have been looking at intro to real analysis. I am using the text book "Principals of Mathematical Analysis, third edition" by Walter Rudin. I have some questions about things I found confusing and ...
2
votes
3answers
82 views

Calculate $\int\frac{dx}{x\sqrt{x^2-2}}$.

The exercise is: Calculate:$$\int\frac{dx}{x\sqrt{x^2-2}}$$ My first approach was: Let $z:=\sqrt{x^2-2}$ then $dx = dz \frac{\sqrt{x^2-2}}{x}$ and $x^2=z^2+2$ $$\int\frac{dx}{x\sqrt{x^2-2}} ...
0
votes
2answers
43 views

Why is this a quotient map

Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...
2
votes
1answer
79 views

If $A=\{ f^{-1}(y)\}$ is uncountable can we find an open interval $I$ such that the set $A$ is dense in $I$?

Let $f\colon [a,b] \to \mathbb{R}$ be a continuous function. Suppose that there is $y$ such that $f^{-1}(y)$ is uncountable. Can it be show that $$\underline{D}f(x)=\liminf_{z\to ...
5
votes
1answer
87 views

A real analytic function that takes each value in $\mathbb{R}$ three times

I was inspired by this question: it is quite easy to prove that for any positive odd number $2m+1$ there exists a function $f\in C^{\infty}(\mathbb{R})$ such that ...
1
vote
1answer
24 views

Bounding the $\ell^{1}$ norm given the $\ell^{2}$ norm

Suppose $x = (x_{1}, x_{2}, \ldots) \in \ell^{2}$. If $\sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1$, is it possible to bound $\sum_{n = 1}^{\infty}|x_{n}|$?
1
vote
1answer
28 views

Prove that a linear and continuous operator admits inverse in Hilbert space

Let $(H,(\cdot,\cdot))$ an Hilbert space and $A:H\rightarrow H$ a linear and continuous operator such that there exists $\alpha >0$ such that $$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$ ...
1
vote
1answer
44 views

Where is “countability” used in this proposition about product $\sigma$-algebra?

The following is a proposition about product sigma algebra from Folland's Real Analysis: Proposition. If $A$ is countable, then $\otimes_{\alpha\in A}M_{\alpha}$ is is the $\sigma$-algebra ...
6
votes
1answer
64 views

How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$ [duplicate]

When $p\geq 2$ and $0\leq x\leq1$, how does one show the inequalities $$ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$$ and $$ 2(1+x^p)\leq (1+x)^p + (1-x)^p \ ?$$ ...
2
votes
1answer
110 views

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times. [duplicate]

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times. Ed.: answered by the duplicate above Does there exist a continuous function ...
1
vote
0answers
27 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
4
votes
2answers
89 views

Bounds of $f(x)=(1-ax)^{1/x}$

Let $a\in(0,1)$ be a fixed number. What is the numeric value of upper and lower bound of $f(x)=(1-ax)^{1/x}$ on $x\in (0,1)$? I feel as though I'm missing something, because it shouldn't be ...
2
votes
2answers
58 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
0
votes
0answers
16 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
0
votes
0answers
13 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
1
vote
1answer
28 views

Find the largest $n\in \Bbb{N}$ answering the following terms

Let $$f(x) = -\frac{1}{12}x^4 + o(x^5)$$ Also, Let $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C & \mbox{if } x=0 \end{cases}$$ I need to find the largest $n\in\Bbb{N}$ ...
0
votes
0answers
89 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
1
vote
2answers
29 views

Borel set, excercise

I need help with this excercise: Prove that every open set is a Borel set by showing that for each open set, $O$, $\chi_O$ is a Borel measurable function. Hint: Begin by showing that $\chi_I$ ...
2
votes
1answer
66 views

How prove $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$

How prove that if $x, y \in (0,\sqrt{\frac{\pi}{2}})$ and $x \neq y$, then $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$?
1
vote
2answers
22 views

extended well ordering property

Suppose $ S=\{m\in \mathbb{Z} | y \lt m \}, y \in \mathbb{R}$ . This set is bounded below by $y$. And it is nonempty since Archimedian Property guarrantes the existence of a natural no $n$ which is ...
0
votes
1answer
29 views

Writing dense sets in terms of set of integers

Can we write every dense set in $\mathbb R$ as {$x_n$}$\mathbb Z$ , where {$x_n$} is a real sequence with limit $0$ ?
1
vote
1answer
61 views

Alternative Uniform-Continuity theorem proof by Luroth

Can please someone elaborately give the proof of Uniform-Continuity theorem ( every continuous function on a closed bounded real interval is uniformly continuous) by Luroth ? thanks in advance
2
votes
1answer
26 views

Countable generation of $\sigma$-algebras

I feel that a positive answer to the following question would be helpful in solving some exercises in introductory measure theory: Suppose $\cal A$ is a collection of subsets of a set $X$ and let ...
1
vote
1answer
17 views

How to show whether a set in a normed vector space is compact or not?

I've tried looking for similar problems but couldn't find any. So here it is. We have a normed vector space $\mathcal{l}_p = \{ x= (x_1, x_2, \cdots) : \sum_{n=1}^\infty |x_n|^p \lt \infty \}$, with ...
2
votes
1answer
45 views

Difference between $\mathcal{E^{\prime}}$ and $\mathcal{D^{\prime}}$

What's the difference between $\mathcal{E^{\prime}}$(the space of compactly supported distributions) and $\mathcal{D^{\prime}}$ (the space of smooth compactly supported distributions)? Examples would ...
2
votes
0answers
42 views

Simplifying an integral involving Gaussian PDF

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty ...
-1
votes
0answers
49 views

Is Intro to Real Analysis Necessary? [closed]

I'm starting a graduate program and was wondering whether Intro to Real Analysis should be taken before Real Analysis. I don't really have much experience with real analysis, but I'm a very ...
3
votes
1answer
78 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
1
vote
0answers
29 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
1
vote
0answers
50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
-1
votes
1answer
41 views

Real analysis question - inequalities and equalities? [closed]

Let $a>0$. Show that if $0<x<y$ then $$x^a<y^a$$
1
vote
0answers
18 views

Characterization of total variation of a complex measure

In the text of Real Analysis by Folland, he defines the total variation of a complex measure $\nu $ as the unique measure $|\nu|$ such that if $\nu = f d\mu $, with $ f$ a $ \mu -$ integrable ...
1
vote
1answer
22 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
1
vote
1answer
41 views

How to check if a given piecewise defined function on $\mathbb R^2$ is a norm?

I want to check if the function $\parallel (x,y)\parallel = \left\{ \begin{array}{cc} \sqrt{x^2+y^2} & \mbox{if } xy \geq 0 \\ \max\{\vert x\vert, \vert y\vert\} & \mbox{if } xy < 0 ...
6
votes
1answer
59 views

Which negative integer powers of 2 belong to the Cantor Set?

Consider the Cantor set $C$, and negative integer powers $2^{-k}$. Clearly, for $k=1$, $2^{-1} \notin C$ since $1/2 \in (1/3, 2/3)$, the first deleted open interval. It is known that $1/4 = ...
4
votes
1answer
48 views

Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
0
votes
1answer
66 views

Prove the uniform convergence of the following function series

Prove that $$\sum_{k=0}^{\infty}\left(1+\frac{k}{x}\right)^{-x}$$ is uniformly convergent on $x\in\left[a,\infty\right).$ According to the equality, $$\frac{x}{1+x}<\ln(1+x)$$ we have, ...
0
votes
0answers
57 views

How to prove a number is not real.

If $a^2 $ = -1, prove that a is not a real number. Can I first assume that a is a real number, then say that a square of a real number is positive and this is contradictory so a is not real?
1
vote
0answers
45 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
1
vote
2answers
55 views

Choosing the right $\delta$ for uniformly continuous function

I'm reading a proof for the claim $g(x)$ is uniformly continuous. It comes down to: $\forall x,y>B:\left|g(x)-g(y)\right|\le \left|x-y\right| + \frac{\varepsilon}{2}$ The auther claims $\delta = ...