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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Don't understand proof that union of open sets is open

All of the proofs I read of this are pretty much the same, but they all don't explain the last part of the proof and end it there because I assume they think it is obvious to everyone. The proofs ...
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1answer
63 views

Proving that the function $\rho$ which sends a lifting of a circle map to its rotation number is continuous.

Let $\mathcal{L}$ denote all circle maps of degree one with nondecreasing liftings (a map $f \in \mathcal{L}$ is of degree one if its lifting $F$ satisfies $F(x+1)=F(x)+1$) . I need to prove that if ...
2
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1answer
57 views

Prove continuous sup-property.

It is a question from pugh's real mathematical analysis. Suppose that $\mathcal E \subset C^0$ is equicontinuous and bounded. (We can consider $C^0$ is the set of continuous functions defined on ...
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1answer
43 views

Stieltjes inversion formula

Let $[a,b] \subset \rho(T)$ and $T$ be a self-adjoint operator then I want to show that $0=\frac{1}{\pi} \lim_{\varepsilon \downarrow 0} \lim_{\delta \downarrow 0} \int_{a+\delta}^{b+\delta} ...
4
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1answer
174 views

Proving a metric space $\mathbb N^{\mathbb N}$ with $d(x,y)=1/\min\{j:x_j\neq y_j\}$ is complete

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
3
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1answer
111 views

How to solve this integration

Is there any easy way to solve this integration? $$\int \frac{3x}{(x^2+x+1)^3}dx$$
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1answer
32 views

For invertible $A$ show that $\lbrace y \in \mathbb{R}^n : \| x-y \|_A < r \rbrace= \lbrace x + A^{-1} y: y \in B_r(0) \rbrace$

I am struggling with the following Problem: Let $| \cdot|$ be the Euclidian Norm on $\mathbb{R}^n$. Let $A$ be an invertible $n \times n$ Matrix. Define $\|x\|_A = |Ax|$ for $x \in \mathbb{R}^n$ ...
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2answers
42 views

norms of operator

I am stuck on a question on the operator norm. If $L$ is a bounded linear operator $L:H\rightarrow H$ where $H$ is a Hilbert space. How would you show that $$\|L\|\leq \sup_{\|u\|=\|v\|=1}(Lu,v)$$ any ...
3
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1answer
106 views

Proof of the spectral theorem

I am currently going to through my proof of the spectral theorem that we had in class, but I feel that I have copied some nonsense from the board. So we defined the Cayley transform $U= ...
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3answers
89 views

How is the fourier series of $\frac{\pi-x}2$ derived?

$$S = \sum_{n=1}^{\infty} \frac{\sin(n)}{n} $$ I seem to have found on the web: $$\frac{\pi-x}{2}=\sum_{n\geq1}\frac{\sin\left(nx\right)}{n} \space, x \in(0, 2\pi)$$ Then: $$x = \pi - ...
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0answers
42 views

Show that for $0<a, b<\infty$ $\int^\infty_0 \frac{\log(|\frac{\tan(bx)}{\tan(ax)}|)}{x}= +\infty \text{ if } b>0$…

Show that for $0<a, b<\infty$ (This is not meaningless it is required for the equation with the $M(f)$ piece) $$ \int^\infty_0 \frac{\ln\Big(\Big|\frac{\tan(bx)}{\tan(ax)}\Big|\Big)}{x} = ...
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2answers
214 views

Suppose $f$ has derivatives of all orders. Prove that $F(x):=exp(f(x))$ also has derivatives of all orders.

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ has derivatives of all orders. Prove that $F(x):=exp(f(x))$ also has derivatives of all orders. Genuinely very confused by this question. I used an ...
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2answers
39 views

Again on an existence of convergent subsequence

Let $(a(m,n))_{m,n \in N}$ be a double sequence of positive numbers. Suppose that we know that there exists a limit $\lim\limits_{m\to \infty}\lim\limits_{n \to \infty}a(m,n)=L$. Does there always ...
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2answers
45 views

A step of change of variables I don't understand when show derivative of a function in $L^n$?

How (16) can goes to (17) and how to show $loglog(1+\frac{1}{|x|})\in L^n$ for $n>1$? My attempt: Let $|x|=r$, then $(x^2)^{\frac{1}{2}}=r$. Differentiate on both sides, I get ...
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1answer
46 views

Using Partial Summation to evaluate a series

$$S = \sum_{x=1}^{\infty} \frac{\sin(x)}{x}$$ Using partial summation. Obviously, $$S = \lim_{n \to \infty} \sum_{x=1}^{n} \frac{\sin(x)}{x}$$ Partial Summation: \begin{align*} \sum_{n=1}^{N} ...
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1answer
66 views

Subsequences and upper and lower limits of a sequence

I'm working on a homework assignment in which I have to find the upper and lower limits of a sequence. I've partitioned the sequence into two subsequences (one consisting of all even terms and ...
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0answers
132 views

sup of integrals of simple functions = inf of integrals of simple functions implies f is measurable?

Let $E \subseteq \mathbb{R}$ be measurable with $|E| < \infty$, and f a nonnegative, bounded function on E. Prove that $sup \lbrace \int_E \phi : 0 \leq \phi \leq f, \phi$ simple$ \rbrace = inf ...
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3answers
105 views

What can you say about the limit of $( x_n)$ as $n$ approaches infinity?

I have a metric space $M$ and a convergent sequence $(x_n)$ in $M$. I also have $a$ in $M$ such that: 1) the set $\{a, x_1, x_2, \dots, x_n,\dotsc\}$ is closed. 2) the set $\{x_1, x_2, \dots, x_n, ...
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2answers
65 views

A question on open sets

If we define open as: A set $O⊆R$ is open if for all points $a∈O$ there exists an $\epsilon$-neighborhood $V_\epsilon(a)⊆O$. Where $V_\epsilon(a) = \{x \in \mathbb{R}: | x - a | < \epsilon\}$ Now ...
3
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1answer
83 views

Updated: Frullani Integral For $0<a , b< \infty$, show $\int^\infty_0 \ \frac{|\sin(bx)|-|\sin(ax)|}{x} dx=\frac{2}{\pi}\ln(\frac{b}{a})$

For $0<a , b< \infty$, show $$\int^\infty_0 \ \frac{|\sin(bx)|-|\sin(ax)|}{x} dx=\frac{2}{\pi}\ln(\frac{b}{a})$$ I this can be written as $$\int^\infty_0 \ ...
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1answer
82 views

Limits of function challenge [closed]

My professor asked me to give an example of a function $f$ defined on real numbers such that $f$ has a limit at $x=5$ only. Could any one help me to find that example.
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0answers
17 views

Prove that outer measure is not finitely additive [duplicate]

In Royden, the author states that outer measure (on the path to constructing the Lebesgue measure) is not finitely additive. Is there a simple proof or counter example that someone can provide of this ...
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1answer
59 views

How to use The Inverse Function Theorem to prove $f$ is a diffeomorphism?

I've proven that the function $f: U=(0,\infty)\times \mathbb R \rightarrow \mathbb R^2$ given by $f(x,y) = (x, y^3 + xy)$ is injective and surjective ($f(U) = U$), so it is bijective. I've computed ...
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2answers
45 views

Prove or disprove: $A\subset B$ $\Rightarrow$ acc$(A)\subset$ acc$(B)$.

Prove or disprove: a. $A\subset B$ $\Rightarrow$ acc$(A)\subset$ acc$(B)$ where acc$(X)$ is the set of accumulation points of the set $X$. b. $\sup A \in \text{acc}(A)$. The answers say both are ...
0
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1answer
41 views

Chain rule multiple variables

Let $u(x,y)=\ln(x+y)e^{\cos(x+y)}$ and $h(x)=u(x,1+x^3)$. What is $\frac {dh}{dx}$? How do I use the chain rule in this case?
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0answers
34 views

Integral with Fresnel functions

$I=\int\sqrt{t}\sin(t)\mbox{d}t$ How to calculate $I$ ? I know, that the answer is $\sqrt{\frac{\pi}{2}}C\left(\sqrt{\frac{2}{\pi}}t\right)-\sqrt{t}\cos(t) +\mbox{const}$, where $C(u)$ is Fresnel ...
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1answer
64 views

Sobolev embedding $W^{1,2}(\Omega)\subset L^p(\Omega)$ where $\Omega$ is a halfplane

I would like to ask when the following Sobolev embedding holds true $$W^{1,2}(\Omega)\subset L^p(\Omega)$$ where $\Omega\subset \mathbb{R}^2$ is any open set and $1 < p < \infty$. All book ...
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0answers
148 views

recommendation for exercises/problem books

I am studying 1.Multivariable Calculus - with emphasis on limits, continuity , differentiability of functions of two variables , maxima minima 2.Linera algebra -emphasis on questions of ...
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3answers
49 views

Show that the function $y \mapsto y^3 + ay$ is injective for $a < 0$?

Show that the map $$y \mapsto y^3 + ay \ \ \ \ \ \ \ (y \in \mathbb R)$$ is injective where $a < 0$. I see that the derivative is given by $\ \ 3y^2 + a$. I've computed $3y^2 + a > 0 \iff y ...
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0answers
26 views

Euler's homogeneous identity

Let $\xi^\nu$ be a unit vector in $\mathbb R^n$, $\psi^\nu$ a smooth function, homogeneous of degree $0$ and we choose coordinates on $\mathbb R^n=\mathbb R\xi^\nu\oplus\xi^{\nu\perp}$ in the ...
3
votes
1answer
65 views

Finding $ \lbrace a_{n}\rbrace $ s.t. $\mathop {\lim }\limits_{n \to \infty }a_{n}=1$ and $\mathop {\lim }\limits_{n \to \infty }a_{n}^{n}=2015$

The following problem appears in our analysis assignment. Find a sequence $ \lbrace a_{n}\rbrace $ of real numbers such that $$\mathop {\lim }\limits_{n \to \infty }a_{n}=1\text{ and }\mathop ...
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1answer
33 views

Prove that a sequence is a Cauchy sequence iff the following holds

Show that a sequence $(x_n)$ in metric space $(S, \rho)$ is Cauchy if and only if $\lim_{n\to \infty}\sup_{k \ge n} \rho(x_n,x_k)=0$ John Stachurski, Economic Dynamics, Exercise 3.2.1
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0answers
12 views

Show $f(x,y)=(x,y^3+xy)$ for $(x,y) \in U = (0,\infty) \times \mathbb R$ is injective and where $y \mapsto y^3 + ay$ is monotonic.

Let $f:U \rightarrow \mathbb R^2$ be given by $f(x,y)=(x,y^3+xy)$ for $(x,y) \in U = (0,\infty) \times \mathbb R$. I want to show $f$ is injective and decide when $y \mapsto y^3+ay$ is monotonic. I ...
4
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2answers
1k views

Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequence ...
3
votes
3answers
87 views

Proof of a limit of a sequence

I want to prove that $$\lim_{n\to\infty} \frac{2n^2+1}{n^2+3n} = 2.$$ Is the following proof valid? Proof $\left|\frac{2n^2+1}{n^2+3n} - 2\right|=\left|\frac{1-6n}{n^2+3n}\right| ...
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1answer
17 views

Is the integral finite if the integrand is $o(x^{-1})$?

According to theorem 2.2 in this file http://www.stat.umn.edu/geyer/old06/5101/notes/n1.pdf If $\lim_{x\to\infty} \frac{g(x)}{x^{-1}} =0$, nothing can be said about the existence of ...
3
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1answer
75 views

Is the set $x^2>2$, $x\in \mathbb{Q}$ both open and closed in $\mathbb{Q}$?

Is the set $S=\{x^2>2$, $x\in \mathbb{Q}\}$ both open and closed in $\mathbb{Q}$? I think yes. My argument: Define open as usual in a metric space. It is clear that $S$ is open. Define closed by ...
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1answer
31 views

Show that $m(\Gamma)=0$, where $\Gamma$ is a curve $y=f(x)$

Suppose $\Gamma$ is a curve $y=f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0$. Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$. If the ...
3
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1answer
42 views

$\sup A \le \beta$ proof verification

If $A \subset B, a_0 \in A$ and $\beta$ is an upper bound of B then $\sup A \le \beta.$ $\textbf{Proof:}$ Since $a_0 \in A,$ then $A \ne \emptyset,$ thus $\sup A$ exists. Since $A \subset B,$ then ...
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3answers
115 views

A sequence for which the set of limits points is the interval $[0,1]$.

My professor challenge me to give a sequence with limit points from zero to one including 0 and 1?
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2answers
96 views

If $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $ f'(x) \leq f(x), \forall x,$ then $f\equiv 0 \ ?$

Suppose $f:\mathbb{R} \to \mathbb{R}$ has two derivatives such that $f(0) =0$ and $ f'(x) \leq f(x), \forall x.$ Could anyone advise me how to prove/disprove that $f \equiv 0 \ ?$ Thank you.
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1answer
352 views

Dual Space Annihilator in C[0,1]

Let $V = C[0,1]$ and let U be the subspace of functions of the form $y(x) = ax+b$ for some a, b depending on the function. Give an explicit family of functionals $F\subset U^\perp$ such that for any ...
0
votes
1answer
87 views

Proving Finite Union of Disjoint Closed Intervals is Closed?

Forgive my poor LaTeX, I'm very new to it (as in, reading guides as I go just to write this). In my Elementary Real Analysis course, we're asked to prove a finite union of closed sets is itself ...
0
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2answers
108 views

Compact Hausdorff space is of second category

Let $X$ be a compact Hausdorff space, prove $X$ is of second category. I found a proof of this theorem in the case of locally compact Hausdorff spaces. Let $E_n$ be open and dense in $X$, locally ...
2
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2answers
151 views

Uniform Continuity: $(\ln x)^2$

Determine if $f(x)=(\ln x)^2$ is uniformly continuous on $(0,\infty)$. I think I need to use the definition on this one. im not sure how tho, so any tips/solutions? $|y-x|< \delta$ => ...
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votes
3answers
116 views

If $a_n>0$ converges to $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a.$ [closed]

I would appreciate your help! How can we show that if a sequence of positive real numbers $a_n$ converges to $a\in\mathbb{R}$ with $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a$. ...
1
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2answers
69 views

Find $\sup f^{n}(x)=\sup \underbrace{f\circ f\circ\cdots\circ f}_{n \text{ times}}(x)$.

Let $f(x)={x\over \sqrt{1+x^2}}$. Find $\sup f^{n}(x)=\sup \underbrace{f\circ f\circ\cdots\circ f}_{n \text{ times}}(x)$. $n\in \Bbb{N}.$ I suppose $\sup f=1$. Isn't $\sup f^{n}(x)=1$ directly? I am ...
2
votes
1answer
41 views

Measure in spectral theorem always positive?

In my functional analysis lecture we introduced the continuous functional calculus on $\sigma(T)$ if $T$ is a self-adjoint operator. Then the Riesz representation theorem gives us that ...
4
votes
2answers
87 views

Let $f({x\over x+1})=x^2$. Find $f(x)$.

Let $f:\Bbb{R}\to \Bbb{R}$ fulfill $f({x\over x+1})=x^2$. Find $f(x)$. I guess there is a rule or a claim I should be using but I can't think o any. It was given in an exercise about ...
0
votes
1answer
73 views

$\langle Tx,x \rangle =0$ then $T=0$

Given a complex Hilbert space $H$, we have that $\langle Tx,x \rangle =0$ then $T=0$ holds. I looked to some old threads and all of them talked about this by referring to the polarization identity, ...