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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Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
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2answers
33 views

$f : ]0, \infty[\to\mathbb{R}$ Prove that $\lim_{x\to\infty} f(x) = 0$ if $\lim_{x\to\infty} [f(x)+f'(x)] = 0$

I tried $ f(x)=\frac {f(x)e^x}{e^x} $ but now I'm stuck. How is it possible to apply the $\frac{\infty}{\infty}$ - LHR?
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1answer
33 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{Nā†’āˆž}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
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1answer
13 views

bounded and connected bounded interval are equivalent [on hold]

X is a subset of real line.Then the following statements are equivalent (a) X is bounded and connected. (b) X is a bounded interval.
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2answers
16 views

The infimum $\inf_{(a,b) \in A\times B} \; \rho(a,b)$ is attained for any two compact sets $A,B$

Let $A,B$ be compact sets in $(S,\rho)$. Define $\rho(A,B)$ by $$\rho(A,B) = \inf_{(a,b) \in A\times B} \; \rho(a,b)$$ Show that there exists $a_0 \in A, b_0 \in B$ s.t. $$\rho(A,B) = \rho(a_0,b_0)$$ ...
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1answer
44 views

Infinity in “Extended Natural Numbers”

In Baby Rudin, p. 27, it is stated that the $\infty$ in notations like $\sum\limits_{i=0}^\infty i$ and $\bigcup\limits_{i=0}^\infty A_i$ is not the same as the $+\infty$ in the extended real ...
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1answer
29 views

Limit point s.t $ S\cap Q $ =$ \emptyset $

Let $S$ be infinite subset of $\mathbb{R}$ s.t $ S\cap \mathbb{Q} =\emptyset $. Which of following statements is true? $S$ must have a limit point which belongs to $\mathbb{Q}$ $S$ must have a limit ...
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1answer
26 views

$Y_n = \sup_{k \geq n} E(X_k | F_n)$ is a martingale if $X_n$ is $L^1$ bounded non-negative submartingale

Let $X_n$ be a $L^1$ bounded non-negative submartingale. Let $Y_n = \sup_{k \geq n} E(X_k | F_n)$. Show that (1) $Y_n$ is a martingale (2) $X_n \leq Y_n$ for all $n$ a.s. (3) $\sup \|X_n\|_1 = ...
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1answer
23 views

How can I compute $\lim_{|r|\to\infty}\int_{\Bbb R}|f(x+r)+f(x)|^p\ dx$ with $f\in L^p$?

This is an exercise in real analysis: Suppose $f\in L^p(\Bbb{R})$ for $1\leq p<\infty$. Compute $$ \lim_{|r|\to\infty}\int_{\Bbb R}|f(x+r)+f(x)|^p\ dx. $$ For $p=1$ and nonnegative ...
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0answers
14 views

L1 norm less than BV norm

I will appreciate any hint on this Prove that if $f$ is a function of bounded variation then $\|f'\|_{L_1} \leq \|f\|_{BV}$. When $f$ is differentiable just by the Fundamental Theorem of ...
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0answers
33 views

Show that function is positive definite

I have been working on this problem for the past couple of days and I am stuck with it. It asks to show that the following function is positive definite $$\alpha(t)=\sum_{k=-\infty}^\infty ...
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2answers
20 views

Continuous functions and open sets

I'm working on a proof and having trouble applying a certain theorem. I want to prove that if $ f $ is a continuous function from a metric space into the real numbers, then the set $ {f(x)>0} $ ...
2
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0answers
29 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that (1) for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ (2) $\|\sup X_n\|_1 ...
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0answers
7 views

On Differentiation Theorem and Radius of Convergence

For reference: http://www0.maths.ox.ac.uk/system/files/coursematerial/2014/2644/48/14MT-AnalysisI-sheet7.pdf 4.(b) For fixed $d \in \mathbb{R}$, let $f_d(x)=S(d+x)C(d-x)+S(d-x)C(d+x)$ By ...
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0answers
17 views

If the support of a function is contained in a Borel set, is the support of its derivative also contained there?

Let $f$ be a function such as $\operatorname{supp}(f)\subset Q$ where $Q$ belongs to the Borel $\sigma$-algebra on $\mathbb{R}^d$ Do we have $\operatorname{supp}(f')\subset Q$?
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2answers
22 views

Why is this line integral not $0$? (An incorrect application?)

I think I am little off my game tonight, but can someone tell me why this integral is not $0$? $$\int_{C} x^2 - y^2 ds$$ where $C$ is an ellipse and $ds = \sqrt{ (dx/dt)^2 + (dy/dt)^2} dt$ To ...
5
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3answers
95 views

Convergence of $ \sum_{n=1} ^\infty \frac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$

Convergence of $$ \sum_{n=1} ^\infty \dfrac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$$ Attempt: I believe not a nice attempt: $ n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} ...
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1answer
22 views

Proof: Cauchy sequences and uniform continuity

I'm working on a proof and I'm having trouble relating definitions I want to prove that if f is uniformly continuous, then if a sequence $ {a_n} $ is Cauchy, $ {f(a_n)} $ is Cauchy. So if $ f $ is ...
2
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3answers
35 views

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$. I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure ...
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0answers
13 views

If $\mu$ is a semifinite measure, every set of infinite measure contains a subset of arbitrarily large, finite measure [duplicate]

I need help in this question. Thank you. If $\mu$ is a semifinite measure and $\mu(E)=\infty$, for $C>0$ there exists $F\subset E$ with $C<\mu(F)<\infty$.
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1answer
9 views

“Multivariable” version of this lemma about showing analytically that a number is irrational.

Lemma: let $\alpha \in \mathbb{R}^+$ and $p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N}$ such that $\left|\alpha q_n - p_n \right| \neq 0$ for all $n \in \mathbb{N}$ and $$ \lim_{n \rightarrow ...
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1answer
29 views

Simple Differential Geometry/Analysis question: Prove that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous

In Differential geometry of curves and surfaces by Manfredo do Carmo, page 459, it says the following: Observe that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous, where $f(x,y) = \frac{x^2}{a^2} - ...
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1answer
18 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
2
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5answers
60 views

Convergence of $\sum_{n=1}^{\infty} \log~ ( n ~\sin \frac {1 }{ n })$

Convergence of $$\sum_{n=1}^{\infty} \log~ ( n ~\sin \dfrac {1 }{ n })$$ Attempt: Initial Check : $\lim_{n \rightarrow \infty } \log~ ( n ~\sin \dfrac {1 }{ n }) = 0$ $\log~ ( n ~\sin \dfrac ...
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17 views

Wieierstrass $M-$ test for uniform convergence

I have a problem regarding Wieierstrass $M-$ test for uniform convergence. Is it necessary to have the hypothesis '$|f_n(x)|\leq M_n$ for all $n\geq 1$'? Or is it sufficient to have it like 'There ...
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1answer
17 views

$\lim_{n \to \infty} \int_{\Omega}X_n d \mu = +\infty$ under some conditions

Suppose $X_n$ are measurable functions in $L^1$ defined on the measure space $(\Omega, \mathfrak{F}, \mu)$. Suppose that $0 \leq X_n$ a.e. for all $n$ and $X_n \leq X_{n+1}$ a.e. for all $n$. Thus ...
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2answers
64 views

Prove that a function is constant [duplicate]

I'm attempting to prove the following statement: Let $f:\mathbb{R}\to\mathbb{R}$ be a function and suppose that $|f(x)-f(y)|\leq (x-y)^2$ for all $x,y\in\mathbb{R}$, therefore f is constant. I ...
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2answers
16 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
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2answers
38 views

Why is $\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \ldots a_n}} = \lim_{n \to +\infty}{a_n}$

Solving some problems regarding limits and sequence convergence, i stumbled upon a task, and it's solution relies on, and i quote: "We now use a well-known theorem : $$\lim_{n \to +\infty ...
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1answer
25 views

Define $x^3$ = $x \times x^2$. Prove that if $x_1, x_2$, . . . represents $x$, then $x_1^3$, $x_2^3$, . . . represents $x^3$ [on hold]

I'm a little bit lost on where to start this problem. My initial thought is to work backwards. Say $x^3$ is a Cauchy sequence. Then for some $j, k \geq m$ contingent on $n$, we have |${x^3}_j - ...
1
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1answer
36 views

Arzela-Ascoli theorem for a set of Riemman Integrable Functions.

Are there conditions which we can impose on a set of Riemann Integrable Functions that are weaker than equi-continuity condition of the original Arzela-Ascoli theorem which still have the result of ...
0
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1answer
23 views

Prove the associative law for the addition of real numbers

The problem asks us to prove the commutative and associative laws for the addition of real numbers. The commutative proof seems straightforward. I am wondering how to approach the proof of the ...
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3answers
56 views

convergence of $ \sum_{n=1}^\infty \frac {1}{\log (1 +\frac {1}{n})}$

Test convergence of $$ \sum_{n=2}^\infty \dfrac {1}{\log (1 +\frac {1}{n})}$$ I am not really sure how to move forward. Could anyone give me a direction to proceed please. EDIT" The only part I ...
5
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4answers
94 views

Prove $d(x,y)=\arctan|x-y|$ is a metric

I have to show that $d$ is a metric on the real numbers, and the first three axioms are straight forward, the triangle inequality poses a problem. I know we need to get $$ \begin{align*} d(x,y) ...
2
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1answer
30 views

right continuous representant of L^1 function

Would someone know if a function in $L^1(\Omega)$ or more generally in $L^p(\Omega)$ must have a right continuous representant. $\Omega$ being a open set of $\mathbb{R}$. Thankfully,
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2answers
47 views

Show that $(\mathcal{M},d)$ is complete metric space

Let $(\Bbb{R},\mathcal{M},\mu)$ be the Lebesgue measure space modulo the equivalence relation $A\sim B$ if $\mu(A\bigtriangleup B)=0$. Let $d(A,B)=\mu(A\bigtriangleup B)$. Show that $(\mathcal{M},d)$ ...
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0answers
34 views

a question concerning infinite product

I have a question about the convergence of infinite product. In real mathematics analysis page 198 question 63, part a) $a_k$=$(-1)^k$/${\sqrt{k}}$, and we need to show series ...
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3answers
63 views

Convergence of $\sum_{n=1}^{\infty} \frac {1}{n\log^2(n+1)}$

Convergence of $$\sum_{n=1}^{\infty} \dfrac {1}{n\log^2(n+1)}$$ Attempt: We note that $\lim_{n\rightarrow \infty} \dfrac {n}{ \log^2(n+1)} = \infty$ Hence, for a sufficiently large $n: \dfrac {n}{ ...
2
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1answer
38 views

Struggling with something 4 sources use “clearly” with (involving $\inf$ of a set)

This is technically measure theory but it as much real analysis. We have a Measure $\mu:R\rightarrow\mathbb{R}_{\ge0}\cup\{\infty\}$ We define the outer measure $\mu^*$ to be: ...
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2answers
26 views

showing a (B,p) is complete

let $(A,d)$ be a metric space which is complete and let $B$ be closed in A. Then prove $(B,p)$ is complete, with $p = d|_{B\times B}$ (i.e. restriction of d onto $B\times B$) thoughts: since $B$ is ...
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40 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
3
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2answers
53 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
2
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1answer
33 views

Polish spaces, closed sets and $G_{\delta}$ sets

In a series of lecture notes regarding descriptive set theory, in the section regarding the Borel hierarchy I found the following statement: We will restrict ourselves from now on to Polish ...
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1answer
29 views

given a metric space, how can I describe converging sequences?

For instance, let $(\mathbb{N},d)$ be a metric space with $ d(x,y)= \dfrac{|x-y|}{1+|x-y|}$, im asked to describe all convergent subsequences for convergent: we knowt $x_n \to x \iff d(x_n,x) \to 0 ...
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2answers
55 views

convergence of $\sum_{n=1}^{\infty} \frac {1}{\log(e^n+e^{-n})}$?

Test convergence of $\sum_{n=1}^{\infty} \dfrac {1}{\log(e^n+e^{-n})}$ Attempt: I have tried the integral test, the comparison test ( for which I couldn't find a suitable comparator). However, I ...
1
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1answer
35 views

Prove $\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}$

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with ...
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1answer
29 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. How one can determine $a, b$ and $p$ such that 1) I ...
0
votes
2answers
25 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
13
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1answer
623 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
1
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0answers
23 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...