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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Prove that if $F$ is an ordered field in which every non empty set which has an upper bound also has a supremum , then $F$ is archimedean

Prove that if $F$ is an ordered field in which every non empty set which has an upper bound also has a supremum , then $F$ is archimedean Attempt: If $F$ is an ordered field, then it possesses a ...
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20 views

Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$ (*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0. $$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
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42 views

Example of limit of a function

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand example 4.2.2 on page 105. The aim of the example is to show that: $$ \lim_{x \to 2} g(x) = 4 $$ ...
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29 views

Are Borel sets preserved by an open continuous map?

Does an open, continuous function defined on a compact metric space to itself send Borel sets to Borel sets?
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18 views

Verification of extension result for Lipschitz functions

does anyone know the following result? If it holds in this form and any source which presents it? Thanks a lot. Consider metric space $(X,d_{X})$. Let $f:A \subset (X,d_{X}) \rightarrow \mathbb{R}$ ...
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2answers
70 views

For what $a \in \mathbb{R}$ does $f_a(x) := \sin x + ax$ attain every value exactly three times?

See the title. This is maybe a twist on a classic calculus question. A reasonable first guess (and indeed my own) would be a value of $a$ such that $f_a(3\pi / 2) = 0$, i.e., $a = \frac{2}{3\pi}$. ...
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43 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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37 views

Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Suppose $f$ is a real-valued (possibly infinite-valued) function on some measure space $(X, \Sigma, \mu)$, and suppose that it is measurable. Note that $f$ is not necessarily nonnegative. Does it ...
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22 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
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8 views

Asymptotic behaviour of mean value vs. ordinary difference

Suppose you have a Hoelder-continuous function $f \colon \mathbb{R} \to \mathbb{R}_{+}$ of any order $\gamma \in (0,1/2)$. I would like to replace the "complicated" difference $\frac{1}{h} ...
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17 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
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33 views

About the domain of points having tangent to a curve

Let the graph of $y=f(x)$ be a curve $C$ and $f''>0$. Prove that if $y_0\leq f(x_0)$ then there exist a tangent of $C$ go through $(x_0,y_0)$ I don't know how to prove the existence.
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56 views

Lower bound for $2\sin^2(y\pi)$

I was trying to understand the proof of a theorem, and the author uses the fact that if $y \in \mathbb{Q} \cap(0, \frac{1}{2}]$, then $$2\sin^2(y\pi) \geq \frac{8}{n^2},$$ where $y=\frac{p}{q}$, ...
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1answer
37 views

Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
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30 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
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1answer
28 views

Function in $L^1$ such that $\lim_{t \rightarrow 0} \int_{B(x, t)} f(s) \; ds \neq 0$?

Is there a function in $L^1$ such that $$\lim_{t \rightarrow 0} \int_{B(x, t)} f(s) \; ds \neq 0?$$ where $x \in \mathbb{R}^n$, $B(x, t)$ denotes the ball of radius $t$.
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29 views

Question about point-wise convergent sequence of functions.

Let $$f_n(x)=n^2x(1-x^2)^n$$ be a sequence of functions on $[0,1]$. For $x=0$ and $x=1,$ clearly $f_n(x)=0$. Also for any $x_0$ in the open interval $(0,1)$, we have $0<1-x_0^2<1$. Therefore $$ ...
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32 views

Estimate for $f^2$ on a Ball from below

Let $f\in C^\infty(B_R(0))$, where $R>0$. For $0<\sigma<1$ require the following properties on $f$: $$ 0\leq f\leq 1,\ f=1 \text{ on } B_{(1-\sigma)R},\ f=0 \text { on }\partial B_R,\ ...
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1answer
32 views

Integrating the two-views of lim sup and lim inf

Preliminaries Let $(x_n)$ be a bounded real number sequence and $ (x_n )_{n≥k} $be a subsequence of $x_n$ which only takes the values of the sequence starting from the k−th term. Let {$x_n $} ...
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1answer
16 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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1answer
33 views

$\lim_{p\rightarrow\infty}||x||_p = ||x||_\infty$ given $||x||_\infty = max(|x_1|,|x_2|)$

I have seen the proof done different ways, but none using the norm definitions provided. Given: $||x||_p = (|x_1|^p+|x_2|^p)^{1/p}$ and $||x||_\infty = max(|x_1|,|x_2|)$ Prove: ...
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2answers
32 views

a custom designed cutoff function whose derivative is bounded above.

I am trying to find a $C^\infty$ function $\phi(t)$ with the following properties. $\phi(t) =1$ for $\lvert t \rvert \le 1$ $\phi(t)$=0 for $t \geq 2$ $\lvert \phi'(t) \rvert \le 2 $ I have tried ...
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2answers
25 views

Subsequence of a sequence converging to its lim sup and lim inf

$ Let (x_n)$ be a bounded real number sequence and $ (x_n )_{n≥k} $be a subsequence of $x_n$ which only takes the values of the sequence starting from the k−th term. Let {$x_n $} and {$x_n$ ...
2
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2answers
67 views

How to show $f(x)$ is bounded?

Consider: $$2x\sin(1/x) - \cos(1/x)$$. How to show $f(x)$ is bounded? Thanks!
2
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1answer
23 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
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1answer
21 views

Analysis inequality of norms problem

This seems to be a bit of an odd one. I have worked out a possible answer, but I have a feeling I am going about this the wrong way. Help would be appreciated. Find $m,M\in \mathbb{R}$ so that for ...
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1answer
46 views

Real Analysis Proof of Upper Bound

It's been a while since I have done any analysis based proofs, and I find myself struggling a good bit. Any detailed help (or examples) would be appreciated. I am given the following facts: 1) For ...
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1answer
24 views

Uncountable subset $S$ of $(0,1) \implies (0,1)$ has subinterval of limit points of $S$?

If $S$ is an uncountable subset of $(0,1)$, is there an interval $(a,b) \subseteq (0,1)$ such that every point in $(a,b)$ is a limit point of $S$?
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2answers
37 views

Closed sets in product topology

I have an assignment, I have to proof that arbitrary product of close sets is closed in the product topology, I think I have to use complements and treat with opens, what do you think?
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31 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
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1answer
37 views

$\|x+y\|$ vs $\|x-y\|$ for reverse triangle inequality

So I am using the text "Elementary Functional Analysis" by MacCluer. In it, for Exercise 1.1, it asks us to prove the Reverse Triangle Inequality (which I have done in the past, using the $x=(x-y)+y$ ...
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0answers
17 views

Trying to find base case in recursive argument that a sequence with only finitely many peak indices has a monotone subsequence.

I'm examining an argument in Fitzpatrick's Advanced Calculus that a sequence $\{a_n\}$ with only finitely many peak indices has a monotone subsequence. His argument is shown below, with a modification ...
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124 views

Inverse of $f(x)=x^n+x$ on $[0,\infty)$

Fix integer $n > 1$. The function $f_n(x) = x^n + x$ is monotone increasing on $[0,\infty)$, and so has an inverse $f_n^{-1}(x)$ that is also monotone increasing on $[0,\infty)$. I'm interested ...
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0answers
35 views

Description of a Space of Functions

Here is my question: Denote by $V$ the following space of functions on $\mathbb{R}$: $f\in V$ if and only if there exists a nonnegative integer $k$, complex numbers $a_1,\ldots,a_k$ and purely ...
2
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1answer
42 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
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14 views

Question concerning restrictions and measures

From Real Analysis, Folland: $\textbf{1.14 Theorem.}$ Let $\mathcal{A} \subset \mathcal{P}(X)$ be an algebra, $\mu_0$ a premeasure on $\mathcal{A}$, and $\mathcal{M}$ the $\sigma$-algebra generated ...
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1answer
35 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
3
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4answers
236 views

Sequence proof (by induction, presumably) giving me trouble.

Let $a_1,...,a_n$ be a sequence of positive numbers. Show that $$(a_1+a_2+\cdots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)\geq n^2$$ Hint: Use the fact that for $x>0$ we ...
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57 views

Prove that every Lebesgue measurable function is equal almost everywhere to a Borel measurable function

Suppose $(\mathbb{R},\Sigma(m),m)$ is our measure space, where $m$ is Lebesgue measure. Also, suppose $f : \mathbb{R} \to [-\infty, \infty]$ is a Lebesgue measurable function. The problem: Prove ...
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1answer
23 views

Arzela-Ascoli and compactness in $C(X), l^p, L^p$

Arzela-Ascoli and compactness in $C(X), l^p, L^p$ $C(X)$ with the uniform norm and $X$ is a compact metric space, a closed and bounded set in $C(X)$ is compact if and only if it is ...
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4answers
54 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
2
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2answers
25 views

If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$

If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$ The proof begins with saying that it is no loss of generality ...
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1answer
25 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
2
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2answers
89 views

Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below

I have read the following (http://www.math.harvard.edu/~elkies/Misc/sol8.html) but I dont understand the last part of the solution: For positive $x<1$, consider the alternating sum $$S(x) = x − ...
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1answer
19 views

How to prove unique adherent value

Let $s_n$ be a real sequence and $m_n=\frac{1}{n+1}\sum_{i=0}^n s_n$. We denote $t_n=\alpha s_n+(1-\alpha)m_n$ where $\alpha>0$. Assume that $$\lim_{n\rightarrow\infty}t_n=l$$ Prove that ...
0
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1answer
19 views

In an Archimedian Field $F$, there is a positive rational element $r$ such that $r < z$ for any $ z>0$ in $F$

In an Archimedian Field $F$, there is a positive rational element $r$ such that $r < z$ for any $ z>0$ in $F$ . Is this statement true? Attempt: An archimedian field $F$ is an ordered field in ...
4
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1answer
63 views

Prove that there exist $a \in [-1,1]$, such that $f'''(a)=3f(1)-3f(-1)-6f'(0)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $a \in [-1,1]$ such that $$ f'''(a) = 3f(1)-3f(-1)-6f'(0)$$ Any hint/idea's how to ...
4
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2answers
116 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
-6
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3answers
31 views

Solving formula with cosine [on hold]

How do I solve for $y$ in the following equation $$d = \frac{1}{60000}\cdot(x - y)\cdot\cos\left(\frac{z\,\pi}{648\cdot10^6}\right)$$
8
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1answer
91 views

Pointwise estimate for a sequence of mollified functions

In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote ... let $\eta_\delta$ be the standard mollifier sequence. Let $u_\delta=\eta_\delta\star u$ and note that for any ...