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.

learn more… | top users | synonyms

0
votes
0answers
58 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 $0<p<1$. ...
0
votes
2answers
112 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
102 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 ...
29
votes
3answers
820 views

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
130 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
91 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
49 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 ...
1
vote
1answer
86 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 ...
6
votes
1answer
134 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
33 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
35 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
51 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
77 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
224 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 ...
3
votes
2answers
120 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 ...
3
votes
2answers
107 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
68 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 $\theta \in \mathbb{R}$?
5
votes
0answers
322 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
1
vote
1answer
30 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 $n\in\Bbb{N}$ ...
1
vote
2answers
65 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 $\chi_I$ ...
1
vote
1answer
77 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
27 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
33 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
75 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
152 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
23 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
66 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 ...
3
votes
0answers
59 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 ...
4
votes
1answer
161 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 ...
2
votes
1answer
88 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
58 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 ...
2
votes
1answer
126 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 $d\nu = f d\mu $, with $ f$ a $ \mu -$ integrable ...
1
vote
1answer
99 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
53 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
90 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
57 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
91 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, ...
1
vote
0answers
80 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
97 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 = ...
0
votes
3answers
71 views

How to show that $a_{n+1} = \frac{a_n^2 +3}{4} $ is increasing.

I'm not good at finding whether a sequence is increasing or decreasing. $a_{n+1} = \dfrac{a_n^2 +3}{4}$ is the recursive sequence where $ a_1 =0$ How to get the approach to do something like this? ...
0
votes
1answer
86 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
3
votes
2answers
106 views

How prove $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?

Let $x\ge 1$. How prove that $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?
4
votes
1answer
100 views

Bounded limit inferior and limit superior of functions on dense sets implies divergence on uncountable dense set.

Reading an article by Akcoglu et al. "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters", which can be found here: ...
8
votes
6answers
343 views

If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges

Let $\displaystyle \sum a_n$ be convergent series of positive terms and set $\displaystyle b_n=\sum_{k=n}^{\infty}a_n$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges. I could ...
1
vote
3answers
80 views

About sequences in metric spaces

Let $(x_{n})_{n\in\mathbb{N}}$ sequence in a metric space $(X,d)$ such that the subsequences $(x_{2n})_{n\in\mathbb{N}}$, $(x_{2n+1})_{n\in\mathbb{N}}$ and $(x_{3n})_{n\in\mathbb{N}}$ are convergents. ...
1
vote
1answer
62 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
2
votes
2answers
365 views

Construction of the completion of a measure space

Let $(X,M, \mu)$ be a measure space. Let $\overline{M}$ be collection of $E \cup Z$ such that $E \in M$ and $Z \subset F$, where $F \in M,$ and $\mu(F) = 0.$ We also know $\bar{\mu}(E) = \mu(E).$ a) ...
1
vote
1answer
190 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
3
votes
3answers
105 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
4
votes
2answers
61 views

Showing $f\in L^\infty$ in a finite measure space

I am studying for a qualifying exam, and I seem to have difficulty working problems involving $L^p$-spaces. An explanation for the following problem would be very helpful! Let $(X, \Sigma, \mu)$ be a ...