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
4 views

How to pick decimal expansion, $(0,1)$ uncountable

Prove $(0,1)$ is uncountable. Suppose $(0,1)$ were countable. List $(0,1)$ as: $x_1=0.a_{11}a_{12}\dots$ $x_2=0.a_{21}a_{22}\dots$ and so on, where $a_{ij}$ are integers from $0$ to $9$. ...
1
vote
1answer
20 views

curves and integral

Find the area between these curves. $$y=\dfrac{3}{2x+1},\qquad y=3x-2;\qquad x=2\quad \text{et} \quad y=0 $$ indeed, I calculate the integral of the blue function between $1$ and $2$. Then, I ...
2
votes
0answers
15 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
0
votes
0answers
15 views

Real analysis-proving limit existence for an evaluated function

I don't understand how to define delta for this problem
0
votes
1answer
26 views

proving a statement based on probability theory

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
0
votes
2answers
27 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
3
votes
1answer
25 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
3
votes
1answer
41 views

length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = ...
2
votes
1answer
61 views

Study the following integral: $\int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x … (\ln^{(k)} x)^s }$

How do I calculate for which values of $s$ the following integral converges? $$\int\limits_{0}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} ...
5
votes
2answers
77 views

Limit of $\frac{f\left(x+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{x}$ as $x\to 0$

I'm trying to answer the following question: Let $f$ be continuously differentiable in all of $\mathbb{R}$ and let $g:\mathbb{R}\to\mathbb{R}$ be a function satisfying ...
1
vote
2answers
30 views

Using the same limit for a second derivative

I've been trying to answer the same question answered here: Second derivative "formula derivation" And I'm stuck in a step that is not addressed both in the answer and in the comments of ...
0
votes
0answers
16 views

Approximating roots

Given $n,r\in\Bbb N$, assume $a=n^\frac{1}r$. Assume that $a_d$ is $a$ truncated to $d$ digits ($d$ is total digits both before and after decimal Eg: truncating $412.243$ to $2$ digits is $410.000$ ...
0
votes
0answers
12 views

diffeomorphism on intervals on R

I came across a line in a proof which involved a diffeomorphism $f:I_1 \rightarrow I_2$ (with $f$ a homeomorphism, $f,f^{-1}\in C^{\infty}$) mapping open intervals in R, which claimed that ...
0
votes
0answers
17 views

Second derivative and Hessian-Matrix

Suppose $f$ is twice differentiable. Why is $Df(a)[v] + \frac12 D^2f(a)[v,a] = \frac 12 \langle H_f(a) v, v \rangle $ ? $Df(a)[v]$ is the multidimensional derivative of $f$ at point $a$ in ...
1
vote
4answers
65 views

Calculate the limit as $x\to0$

I need to calculate the limits as $x$ tends to $0$ For the first one, I get that the limit is zero, by splitting it up into $x^3(\sin(1/x))$ and $x^3(\sin^2(x))$ and using the sandwich theorem on ...
2
votes
1answer
18 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
6
votes
1answer
47 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
0
votes
1answer
28 views

metric space: equivalence of several mertric.

I have two questions: Q1) Are all metric on a metric space are equivalent ? Q2) If not: Let $d_1,d_2$ two metric on $X$. If something has a property with a $d_1$ will it hold for $d_2$ too ? For ...
1
vote
2answers
166 views

Proving the equation $C=2x/(1+x^2)^2$ has two positive solutions

I want to show the equation $C=\frac{2x}{(1+x^2)^2}$ has two solutions where $C$ is a constant such that $0<C<\frac{1}{2}$. My method was to say at $x=1$, $RHS=\frac{1}{2}>C$ and I'm sure ...
0
votes
1answer
19 views

Map between two metric spaces and their limit

I have to proof the following: Let $f: V \to W$ be a map between two metric spaces. Proof that $f(a)$ with $a \in V$ is continuous if and only if, for every converging sequence $x_n$ in V with limit ...
1
vote
3answers
57 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
1
vote
1answer
17 views

if one of the sets A and B is compact then d(A,B)>0.

Let $A$ and $B$ be two nonempty disjoint subsets of $\mathbb{R}^{n}$. Put $d(A,B)=inf\left \{ ||a-b||:a\in A, b\in B \right \}$. a) Show that if one of the sets $A$ and $B$ is compact then ...
2
votes
1answer
27 views

Let $\mathcal R$ be a $\sigma$-ring, then: $\{E\subset X; E\cap F \in \mathcal R\text{ for every } F \in \mathcal R\}$ is a $\sigma$-algebra.

I'm trying to Solve the following question: Let $X$ be a non empty set and $\mathcal R$ be a $\sigma$-ring from subset of $X$. Prove that: $$S=\{E\subset X; E\cap F \in \mathcal R\text{ for ...
0
votes
1answer
13 views

Finding the upper boundary of a sequence to find it's limit

I have to determine if the following sequences converge and if they converge I have to determine to what they converge and proof this. $$ a_n = 2^{-n} \\ b_n = \frac{n^2}{n^3 -10} \\ c_n = 1 ...
0
votes
2answers
19 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
0
votes
0answers
10 views

Infinite-dimensional Rieman Integral and Monte Carlo method

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\xi_k)_{k \in N}$ be a sequence of independent uniformly distributed on the interval $[0,1]$ real valued random variables such that $0 \le ...
-1
votes
1answer
19 views

Check differentiablity of $f$ [on hold]

Consider a function \begin{equation*} f(x)=|\cos x|+|\sin (2-x)|. \end{equation*} At which of the following points f is not differentiable? a)$\{(2n+1)\frac\pi2:n\in \Bbb Z\}$ b)$\{n\pi:n\in \Bbb ...
0
votes
1answer
20 views

Determine whether the limit exists and justify answer

Usually for these types of questions, I use sequences of functions to show that the limit does not exist, but I don't think I can do this here? I feel like the limit should be zero, but I don't know ...
-2
votes
1answer
42 views

Existence of divergent series $\sum_{n=1}^ \infty a_n$ of real numbers whose partial sums are bounded and $\lim (na_n)=0$ [on hold]

Does there exist a sequence $(a_n)$ of real numbers such that $\lim_{n \to \infty} (na_n)=0$ , the partial sums of $\sum_{n=1}^ \infty a_n$ are bounded , but $\sum_{n=1}^ \infty a_n$ is divergent ? ...
0
votes
0answers
16 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
4
votes
1answer
74 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
1
vote
1answer
20 views

Characteristic Function in the subset E

Let $E \subseteq \mathbb R$. Then the characteristic function $\chi_{E}:\mathbb R \to \mathbb R$ is continuous if and only if a) $E$ is closed. b) $E$ is Open. c) $E$ is both Open and Closed. d) ...
1
vote
1answer
28 views

What is the graphical representation of Lipschitz continuity?

I know the graphical represntation for Continuity, Uniform continuity,Absolutely continuity, but I want to know the graphical representation of Lipschitz continuity.
4
votes
1answer
23 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
2
votes
1answer
21 views

Counterexample to the double integral computational theorem when the double integral existence assumption is dropped?

To make things simple, consider the simplest case of the double integral computational theorem. Throughout any phrase involving integrability is in the Riemannian sense. Let $[a, b], [c, d] \subset ...
0
votes
1answer
21 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
0
votes
0answers
15 views

Is it possible to modify norm of Sobolev space suitable for ill-posed problems.

I have trying to pose my problem mathematically for quite some time now. I am not sure even if I am close to defining properly. Would anyone please help: I am interested to study ill-posed problem of ...
2
votes
1answer
81 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
1
vote
1answer
23 views

Is the function $f(x)=x^2$ absolutely continuous on the real line?

In Wiki (Lipschitz), it says: A Lipschitz function $g : \mathbb{R}\to \mathbb{R}$ is absolutely continuous. According to the definition of absolute continuity, I am confused about an simple ...
2
votes
6answers
100 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
-3
votes
1answer
35 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
2
votes
0answers
39 views

$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for nonzero rational coefficients

is it possible to find $a,b\in\mathbb{Z}$ such that for every $k\in\mathbb{N}$ the sum $$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $$ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for some nonzero ...
3
votes
3answers
112 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
6
votes
1answer
62 views

the series: compute $ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $

Compute $$ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $$ the result is $\frac{\pi^4+30\pi^2-384}{768}$, so I'm sure the sums $\sum\frac{1}{n^2}$ and $\sum\frac{1}{n^4}$ should appear in the solution. ...
2
votes
1answer
26 views

If $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$

Is it true that if $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$? $m_k$ and $n_k$ are integers.
0
votes
0answers
22 views

Help in finding a paper on nonlinear Schrodinger equations [on hold]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
1
vote
2answers
56 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
0
votes
0answers
24 views

Proof of $f(x+c) \rightarrow m$ as $x \rightarrow a$

If $f(x) \rightarrow m$ as $x \rightarrow a+c$ then $f(x+c) \rightarrow m$ as $x \rightarrow a$. Proof: Because of the assumption, we have $\forall \varepsilon >0$, there exists $\delta >0$ ...
1
vote
2answers
28 views

Is this an open set in $\mathbb R^2$?

Is $\{(x,y)\mid y = \sin \frac {1}{x}, x>0\}$ an open set? (It is living in $\mathbb R^2$.) I think it should be open because $(0,0)$ seems to be a limit point of this set while it is not an ...
4
votes
5answers
89 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$