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 $f$ is Riemann integrable

Let $\{b_n\}$ be decreasing sequence converging to $0$. Define $f$ in terms of $b_n$ as follows $f(0)=1$ and $f(x)=0$ if $x$ is irrational, and $f(\frac{m}{n})=b_n$ if $x=\frac{m}{n}$ with ...
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1answer
49 views

I dont understand how this Maclaurin series got manipulated into looking like this other Maclaurin series. Help Please

I have been reading a book on approximating $e$ and there is a couple lines that I am stuck on. Here they are: $x$ln$(1+\displaystyle\frac{1}{x}) = 1 - \displaystyle\frac{1}{2x} + ...
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10 views

Show that I is a non-empty interval if every continuous function has an interval as its image

Assuming $I$ is a non-empty interval of real numbers I want to show that for any continuous function $f$ that $f(I)$ is also an interval. So given, say, $y_{1}, y_{2}$ in $f(E)$ and without loss of ...
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1answer
45 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
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48 views

Composition operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to H^{-1}(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. Of course, $g(0) = 0$. I believe that $g \in ...
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2answers
44 views

Computation of a sum $S(n)$

We have the following sum: $ \forall \; n \in \mathbb{N} \setminus \{0, 1 \} $ we define: \begin{equation*} \begin{split} S(n) & = (1 \times 2) + (1 \times 3) + \dots + (1 \times n) \\ & + ...
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2answers
55 views

continuity and limit of a function.

Below is the question: To what degree would the sequence definition of continuity need to be modified in order to be suitable as a definition for the limit of a function? In other words,if ...
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29 views

Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
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1answer
45 views

rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of ...
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2answers
45 views

Prove by induction (real analysis exercise)

it would be great if I could get some help with this one! Prove by induction that: $x_n > 2$ for all $n \in \mathbb{N}$, where $x_1 = 5$ and $x_{n+1} = \frac{1}{2}$($\frac{4}{x_n} + x_n)$ ...
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4answers
28 views

Is functions of cauchy sequences is also Cauchy?

Recently i saw in some book that if a sequence is Cauchy then function of that sequence is also Cauchy.I have confusion about this. Please help me.
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1answer
160 views

True or False. Convergent subsequence

Is the statement true or false? If $(x_n)$ has a convergent subsequence,then $(x_n)$ is bounded. The statement is False. However, can someone please show me an example of a sequence with ...
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0answers
32 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
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1answer
18 views

THE sigma-ring or A sigma-ring?

I have two questions about sigma-rings and measure spaces. Let $(\Omega, \mathscr{F}, \mu)$ be any measure space. Is $\mathscr{F}$ THE sigma-ring of this space or A sigma-ring of this space? If ...
2
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1answer
22 views

Measure theory: proof of the “Standardproof” given theorem.

Measure theory: proof of the "Standardproof" given theorem. Let $(X, \mathcal E)$ be a measurable space. Let $W \subseteq \mathcal M(\mathcal E)$ (set of measurable $\mathcal E$-$\mathcal ...
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3answers
34 views

sequence consisting of finite number of distinct values.

This is a question from my previous year analysis exam .The question says : Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number ...
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2answers
15 views

Sequence converging to one.

Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows. Can someone please help me? I don't know what to assume ...
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1answer
36 views

Appoximation of Lipschitz functions by $C^1-$functions

I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
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1answer
30 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
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20 views

Question concerning what a certain notation means. Its usually in calculus

$x$ln$(1 + \frac{1}{x}) = 1 +$ln$(1 + \displaystyle\sum_{i=1}^n \displaystyle\frac{a_{i}}{x^i}) + O(x^{-n-1})$ for $x \rightarrow \infty$ and $n \in \mathbb{N}$. My question is as follows:I have seen ...
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1answer
17 views

Question about pointwise convergence of sequences in the box and product topologies.

Can someone please verify my proof or offer suggestions for improvement? I'm aware that there may be answers floating elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is ...
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33 views

function differentiable but not continuously differentiable

Can I found an example of a function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ such that $f$ differentiable but it is not continuously differentiable and it is not invertible on a point??? Any idea? ...
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25 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...
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1answer
20 views

Decreasing sequence, bounded below.

Suppose that $0 \leq x_1 < 1$ and $x_{n+1} = 1 - \sqrt{1 - x_n}$ for all natural $n$. Prove that $x_n$ is decreasing and bounded below as $n$ converges. Attempt: Suppose that $0 \leq x_1 < 1$ ...
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1answer
29 views

Does the sequence converge or diverge?

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) ...
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1answer
18 views

If $a$ is a real number greater than one and $x$ and $y$ are rationals with $x\leq y$ then $a^x\leq a^y$

How can I show the next proposition?: If $a$ is a real number greater than one and $x$ and $y$ are rationals with $x\leq y$ then $a^x\leq a^y$ Please, I don't know how to proceed, any help ...
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4answers
62 views

Proof that the set of irrational numbers is dense in reals

I'm being asked to prove that the set of irrational number is dense in the real numbers. While I do understand the general idea of the proof: Given an interval (x,y) choose a positive rational ...
2
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1answer
25 views

Prove comparison test for infinite series; Munkres

Prove the comparison test for infinite series: If $\vert a_i\vert \leq b_i$ for each $i$, and if the series $\Sigma b_i$ converges, then the series $\Sigma a_i$ converges. [Hint: show that the series ...
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22 views

Correctness of Proof that the limit of $\sqrt n\cdot c^n$ as $n$ tends to $\infty$ is $0$

The problem is: Given that $|c|<1$ prove that $\lim_{n\to\infty} \sqrt n \cdot c^n =0$. I am asked to use the comparison lemma and archimedean property to show convergence for sequence $\{1/\sqrt ...
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1answer
25 views

Proof that a set that has supremum property implies it has infimum property

I really have no clue where to start with this proof: Suppose S is an ordered field and satisfies supremum property. Show that S has the infimum property as well. Any help would be great.
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22 views

Correctness of Proof that the Archimedean Property of Reals is equivalent to lim $1/n$ as n tends to infinity

Here's what I have gotten so far: The Archimedean Property states 1) For every $\epsilon$ >0 there is a positive integer n s.t. $1/n$< $\epsilon$ and 2) For every positive number c there is a ...
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58 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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2answers
26 views

Proving a set identity

If $A \subseteq X$ and $A_{\alpha}$ is a collection of all such subsets, I need to prove that: $$\left(\bigcap_{\text{all }\alpha} A_\alpha\right)^c = \bigcup_{\text{all }\alpha} A_{\alpha}^c$$ My ...
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30 views

About counting measure on Borel sets

Let $\mathcal{C}$ be all finite unions of half open intervals, $\mathcal{A}=\sigma(\mathcal{C})$, i.e., the Borel $\sigma$-algebra. Suppose that $\mu$ is the counting measure, and $\nu=2\mu$. Can ...
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1answer
21 views

Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
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1answer
42 views

Dimension for a closed subspace of $C[0,1]$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.
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40 views

For a compact set $K\subset \Bbb R^n $ prove the following :

For a compact set $K\subset \Bbb R^n $ and $\delta>0$ show that that there exists a finite number of elements in $K$, say $x_1,x_2,\dots,x_k$ such that any other element $x$ of $K$ is at a ...
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1answer
25 views

Critical points characterization of real function

Providing a real multi-variable function $f(\bar{x})$ twice differentiable with respect of all its variables. Looking for critical points is equivalent to solve $\nabla f = \vec{0}$. And to ...
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0answers
15 views

Integrability in bounded set

Let $A$ be a bounded open set in $\mathbb R^n$; let $f:\mathbb R^n \to \mathbb R$ be a bounded continuous function. Give an example where $\int_\bar A f$ exists but $\int_A f$ does not. Is about is ...
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1answer
19 views

Adding integrals with different domains

Suppose I have two integrals $$ \int_{\Omega_1} f \, \, d \eta$$ and $$ \int_{\Omega_2} g \, \, d \eta$$ how would I define the sum of these two integrals? Is it possible? I want something of the ...
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borel sigma algebra and closed rectangles [closed]

Let $E$ be the collection of closed rectangles in $\mathbb{R}^{d}$. a)Show that $E$ generates the Borel $σ$-algebra on $\mathbb{R}^{d}$ ? b) Show that Lebesgue measure $m$ is the only measure on ...
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+100

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_1$. Then, show that ...
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1answer
26 views

If $f$ is $C^1(\mathbb{R})$, is it $C^1(\{a\})$?

Say I have a well-behaved function like $f(x)=x$. This is obviously $C^1$, but does it make sense to say the function is $C^1$ around a single point? A broader question, if $a\in\mathbb{R}$, does ...
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1answer
54 views

Two reasons why $\int^{1}_{0}f(x) \,dx$ exists?

Consider $f$ on $[0,1]$ defined as $f(0)=0$ $$f(x)=2^{-n}\quad \text{if}\quad 2^{-n-1}<x\le2^{-n},$$ for $n=0,1,2,3,...$ I'm looking for two reasons why $\int^{1}_{0}f(x) \,dx$ exists? One ...
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1answer
40 views

Proof of limit of sum of series

I know that $$\exp(-x)=\sum_{n=0}^\infty\frac{(-1)^{n+2}x^n}{n!}$$ converges for every positive $x$ Then we should have $$\lim_{x\rightarrow+\infty}\sum_{n=0}^\infty\frac{(-1)^{n+2}x^n}{n!}=0$$ ...
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1answer
106 views

Limit of the sum of $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t$

Let $f$ be a continuous, decreasing function, with $\displaystyle\lim_{x\rightarrow\infty}f(x)=0$. Let $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t,\displaystyle x>0$. Let ...
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0answers
89 views

Is it possible to determine the value of the following function?

Let $a_n$ and $b_n$ be a pair of generic sequences, and let $L$ be a constant. Consider the following function: $$f(x):=a_o+\sum_{n=1}^{\infty}\left({a_n}\cos{\frac{n\pi x}{L}+b_n\sin{\frac{n\pi ...
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1answer
29 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
3
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1answer
54 views

Absolutely continuous function, possible?

Is this possible? $\{f_n(t)\},n\geq1, t\in[0,1],$ is a sequence of absolutely continuous functions with $f_n(0)=f_n(1)=0.$ $$\int_0^1f_n'(t)^2dt\leq C<\infty,$$ but ...
0
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0answers
51 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...