Tagged Questions

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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5
votes
6answers
2k views

Archimedean property concept

I want to know what the "big deal" about the Archimedean property is. Abbott states it is an important fact about how $\Bbb Q$ fits inside $\Bbb R.$ First, I want to know if the following statements ...
5
votes
1answer
558 views

Interior of a convex set is convex [duplicate]

A set $S$ in $\mathbb{R}^n$ is convex if for every pair of points $x,y$ in $S$ and every real $\theta$ where $0 < \theta < 1$, we have $\theta x + (1- \theta) y \in S$. I'm trying to show that ...
14
votes
2answers
2k views

A “non-trivial” example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it ...
2
votes
1answer
118 views

$L^p$ derivative vs normal derivative.

Let $f, g : \mathbb{R} \rightarrow \mathbb{C}$ be Lebesgue measurable functions, and let $1 \leq p < \infty$. If $f, g \in L^p$ and $$ \large \lim_{y \rightarrow 0} \normalsize \left\| ...
1
vote
1answer
480 views

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that ...
2
votes
1answer
78 views

Is the inclusion map in the Sobolev embedding theorem a surjective map?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of all real valued functions on $\mathbb{R}^n$ whose first $k$ weak derivatives are in $L^p(\mathbb{R}^n)$. Assume that $$ \frac{1}{q} = \frac{1}{p} ...
3
votes
2answers
121 views

Question on Rudin's Proof of $\int_{-n}^{n}\hat f(\xi)e^{it\xi}\, d\xi\to f$ in the 2-norm

In Real and Complex Analysis pg. 187, (d) property of Theorem 9.13 says: If $f\in L^2(\mathbb R)$ then: \begin{equation}\int_{-n}^{n}f(t)e^{-it\xi}\, dt\to \hat f\text{ and }\int_{-n}^{n}\hat ...
2
votes
1answer
77 views

Local constant interpolation in $L^1$

I really hope that anybody of you can help me with the following question: Consider the set $U\subseteq L^1([0,1])$ of non-negative integrable functions with unit mass, i.e. $u\geq 0$, $\int_0^1 u\,dx ...
1
vote
2answers
65 views

Coefficients of series

Suppose that i have a function $f(x)=\sum_{i=0}^{\infty}a_ix^i$ with radius of convergence $r_f>0$ and that i want to write $f$ in a form $f(x)={e^{g(x)}}$ where $e$ is natural logarithm base and ...
4
votes
3answers
175 views

Convergence of $\sum\limits_{n=3}^\infty \frac1{(\log(\log n))^{\log n}}$

I can prove the sum $\sum\limits_{n=3}^\infty \dfrac1{(\log(\log n))^{\log n}}$ converges this way: I assume that $\exists n_0 \in \mathbb{N}$ such that $\forall n, n \ge n_0$ we have that ...
1
vote
1answer
37 views

What is the basis for $L^2(\Bbb R)$ ? ($\Bbb R$ is the real field)

I know the basis of $L^2(D)$ if $D$ is connected closed interval. But i do not know this.
2
votes
0answers
44 views

Showing $\frac{d}{dx}\left(\frac{f(x)}{1 + cf(x)}\right) \rightarrow 0$ as $c \rightarrow \infty$

The problem I am working on is as follows: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuously differentiable, periodic of period 1, and nonnegative. Show that ...
2
votes
1answer
55 views

Is there a theorem like this in analysis?

If $f$ is continuous on $(a,b)$ (possibly at the end points as well), then $F(x) = \int_{a}^{x} f(x)dx$ is also continuous for some $x \in (a,b)$. I know I am missing some other assumptions that ...
1
vote
1answer
247 views

The Riemann Integrability of a function similar to Dirichlet's function

$\textbf{Problem:}$ Consider the function $f: [0,1] \rightarrow \mathbb{R}$ defined by letting $f(x)=0$ for rational $x$ and $f(x)=x$ for irrational $x$. Calculate the upper and lower Riemann ...
1
vote
2answers
89 views

What duality can you quote that says supremum always exists $\implies$ infimum always exists of a bounded set?

Say you've proven that for a subset of the reals bounded above, there exists a supremum of the set in the reals. How do you prove the dual version for infimum without going through all the steps ...
1
vote
1answer
41 views

Power of exponentials functions

A follow-up question to this question and this comment. Define the set of functions $(f_m)$ for $m\in\mathbb{Z}$ where $f:\mathbb{R}\to(0,\infty)$ is given by $$f_m(x)=\exp(m x)$$ How is it possible ...
3
votes
2answers
53 views

Shifted exponentials functions

Define the set of functions $(f_m)$ for $m\in\mathbb{Z}$ where $f:\mathbb{R}\to(0,\infty)$ is given by $$f_m(x)=\exp(x+m)$$ How is it possible to prove that the functions $f_m$ are linearly ...
0
votes
1answer
261 views

continuous bounded function on the bounded open interval ]a,b[ that is not uniformly continuous on ]a,b[

Does someone know a continuous bounded function on the bounded open interval $]a,b[$ that is not uniformly continuous on $]a,b[$. Thanks in advance
3
votes
2answers
147 views

Why is this set compact?

Let $x_n$ converge in a metric space to $x$, and $F_n=\{x, x_n,x_{n+1}, x_{n+1}, ...\}$. Why is $F_n$ compact? I was going to invoke the following theorem "A subset $A$ of a metric space is compact ...
1
vote
1answer
100 views

Hahn Banach theorem with no dominating sublinear functional

Let $V$ be a vector space and $M$ be subspace of it. If $f$ is a linear functional on $M$, is it possible to extend it to the whole space $V$? If we have a sublinear functional $p$ on $V$ dominating ...
7
votes
2answers
620 views

How to prove $\lim\limits_{n\to\infty} (n+1)^{1/n} = 1$

We know that $\lim\limits_{n\to\infty}n^{1/n} = 1$. Using this, how can we prove that $\lim\limits_{n\to\infty} (n+1)^{1/n} = 1$? Recalling the proof of the former limit, I was able to modify it to ...
1
vote
1answer
234 views

Extension of a bounded and uniformly continuous function

I read in a book the following assertion: Let $\Omega\subset\mathbb{R^n}$ be a open set, then any bounded and uniformly continuous function in $\Omega$ has a unique bounded continuous extension to ...
1
vote
2answers
77 views

When is $\lim_{r\to 0}\int_{-K}^K f(rx)dx=\int_{-K}^K \lim_{r\to 0} f(rx)dx$ true?

When is this true? $$\lim_{r\to 0}\int_{-K}^K f(rx)dx=\int_{-K}^K \lim_{r\to 0} f(rx)dx$$ Is it true without the hypothesis of continuity of $f$? Thank you.
0
votes
0answers
22 views

On an equality of $x \bmod 1$ (the non-diagonal case)

Let $x \bmod 1\in \Bbb R$ be fractional part of $x \in \Bbb R$. For what pairs of $(x,y) \in \Bbb R^2$ with $x\neq y$ is $x^2 \bmod 1 = (y \bmod 1)^2$? The case of $x=y$ was answered in On an ...
5
votes
1answer
203 views

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
2
votes
1answer
40 views

On an equality of $x \bmod 1$

Let $x \bmod 1\in \Bbb R$ be fractional part of $x \in \Bbb R$. For what values of $x \in \Bbb R$ is $x^2 \bmod 1 = (x \bmod 1)^2$? Naturally if $|x| < 1$ we have the equality. However, there are ...
1
vote
1answer
73 views

Why is $\limsup_{n\rightarrow \infty} (1/n) \log(a_n) < 0 \Rightarrow \sum_{n=0}^{\infty} a_n < \infty$ true?

Let $(a_n)$ be a real values series. Why is the following implication true? $$\limsup_{n\rightarrow \infty} (1/n) \log(a_n) < 0 \Rightarrow \sum_{n=0}^{\infty} a_n < \infty$$
0
votes
1answer
143 views

What are measurable sets?

Let $(X \times R,m\times m', \mu \times \lambda)$ be a measure space. If $E$ is any measurable set in $X \times R$ and if $\alpha,\beta \in R$ s.t $\alpha>0$, then $\{(x,y)~:~ (x,\alpha ...
1
vote
1answer
42 views

On showing a distribution is a function

Consider the distributional equation $$\Delta \omega-\omega=\mu$$ Then it is easy to verify by Fourier transform that $$\omega=-\mathcal{F}^{-1}\left(\frac{1}{|\cdot|^2+1}\hat{\mu}\right)$$ is the ...
0
votes
0answers
36 views

A question on $x \pmod 1 \in \Bbb R^n$?

Let $x \pmod 1 \in \Bbb R^n$ be the fractional portion of $x \in \Bbb R^n$. With $A\in\Bbb R^{n\times n}$ as orthogonal and $S(x)$ as sum of coordinates of $x$, for which $x\in\Bbb R^n$ does $S(Ax ...
4
votes
2answers
116 views

Am I right in my conclusions about these series?

I'm trying to decide if these series converge or diverge: $$\sum_{n=1}^{\infty} (-1)^n \left(\frac{2n + 100 }{3n + 1 }\right)^n $$ Here $\lim_{n\to\infty} \left(\frac{2n + 100 }{3n + 1 }\right)^n ...
4
votes
2answers
80 views

How to decide convergence or otherwise of these series?

How to decide whether the following three series converge or diverge? $$\sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n} $$ By the limit comparison with the divergent series ...
6
votes
2answers
113 views

Given that $5a+2b+3c=10$, What is the minimum value of $a^2+b^2+c^2$?

The question is , Given that $$5a+2b+3c=10$$ What is the minimum value of $$a^2+b^2+c^2$$? I know that I have to use AM-GM inequality somehow but I have no idea how to use it for this problem. Help ...
2
votes
1answer
56 views

Integral of $g(x)=1/x$

Let the function: $$ g(x)= \begin{cases} 1/x & x\neq0 \\ 0 & x=0 \end{cases} $$ Let $g^+(x)$ be the positive part of $g$ and $g^-(x)$ the negative part. Then suppose that ...
2
votes
2answers
103 views

On the existence of a particular solution for an ODE

The problem asks to find a bounded $u(\cdot) \in \mathcal{C}^2(\mathbb{R})$ such that $$u''+u'-2u=f$$ where $f$ is a bounded continuous function on the real line. [Observations, Editted] We can ...
2
votes
5answers
194 views

Continuity on $\Bbb R$

$f:\Bbb R\to\Bbb R$ is continuous on $\Bbb R$. $\lim\limits_{x\to\infty}f(x)=0$, and $\lim\limits_{x\to-\infty}f(x)=0$. Prove that $f$ is bounded on $\Bbb R$ and attains either an absolute maximum ...
1
vote
2answers
355 views

A basic question in the definition of limit point

For any subset of $R$ with the usual distance metric, any point inside it is a limit point. Only when the set is discrete there may be a point inside it which is not a limit point. Is this correct ? ...
1
vote
1answer
77 views

Is this proof regarding boundary and closed sets correct?

I've tried to prove the following: Let $A\subset \mathbb{R}^n$, then $A$ is closed if and only if $\partial A\subset A$. My proof is as follows: Notice first that we always have $\partial A \subset ...
1
vote
2answers
195 views

A quadratic form is continuous on $\mathbb R^n$

Prove that the following quadratic function is continuous in $\mathbb{R}^{n}$, ...
4
votes
1answer
113 views

Finding a common subsequence given two sequences on the same elements

Let $A = \{x_1, x_2, x_3,\dots\}$. Claim: If a sequence $y_k$ in $A$ contains infinitely many distinct elements, then it has a subsequence $\{y_{k_m}\}$ that is a subsequence of the $x_n$ ...
0
votes
1answer
143 views

let $A$ a nonempty subset of $ \mathbb{R} $ that is bounded above, then $ \bigcup A \in A $??

in this question $ \mathbb{R} $ is defined as set of Dedekind cuts on $ \mathbb{Q} $ let $ A $ a nonempty subset of $ \mathbb{R} $ that is bounded above, and $ \bigcup A:=\{x|\exists B \in A(x \in ...
1
vote
1answer
56 views

Continuity of generalized mean functions

I'm studying generalized mean functions, and somewhere I found that a weighted mean function could be defined as $M: (0,\infty)^n \rightarrow (0,\infty)$ with the properties: Fixed Point : ...
4
votes
4answers
2k views

Supremum and Infimum of this set..

Am I correct in taking the first few values for $m$ and $n$ as $1,2$ then $2,3$ then $3,4$ respectively? How do I go about finding the supremum and infimum for this sequence? Also how do I ...
2
votes
1answer
212 views

Every countable closed set in $\mathbb{R}^{k}$ has isolated points.

The above statement is the Corollary to exercise 28 of chapter 2 in Baby Rudin. It seems to me like the Cantor set is an obvious exception to this statement. What am I missing?
4
votes
0answers
38 views

On the existence of a weight function making sequence of integration preserving limit

The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there ...
0
votes
2answers
59 views

Find lim of the sequence

Let $(a_n)$ be a sequence where $a_1 \in \mathbb{R} $ and $a_{n+1}=\left | a_n-2^{1-n} \right |,\forall n\in \mathbb{N}^*$ Find $\lim_{n \to +\infty }{a_n}$
0
votes
1answer
73 views

Performing Dedekind cuts on hyperreals

It is a well-known fact that the real numbers are the only complete totally ordered field. So, if we perform the Dedekind cut construction on the hyperreals, then the result must either be: The real ...
2
votes
1answer
92 views

Curious function problem (EDIT: Not so curious, but didn`t see it at the time of writing)

This one is directly from my head and although it could be something trivial I do not see the way to attack it but the problem looks interesting and I want to share it with you, here it is: Let us ...
3
votes
1answer
87 views

What is the sum of the series?

How to find the sum of the series $$\sum_{n=0}^\infty \frac {x^{3^n}+(x^{3^n})^2} {1-x^{3^{n+1}}}$$ under the assumptions $x >0,\,x\neq 1,$ in a closed form?
0
votes
2answers
81 views

Finding the maximum value of $t + \frac{1}{t}$

Finding the maximum of $t + \frac{1}{t}$ by making the first derivative equal to zero and taking $t=-1$ (because second derivative < 0) gives -2 as the answer. But this is not correct. I guess I am ...