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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1answer
36 views

Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. [duplicate]

Define $A(K) = \{f : K \rightarrow \mathbb{R}$ $| f$ is continuous$\}$. $K$ is compact. Prove that $(A(K), ||$ $||_{\infty})$ is a Banach space. Since a Banach space is complete then every Cauchy ...
2
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2answers
41 views

If L = lim_x→a f(x) exists, then |f(x)| → |L| as x → a .

Suppose that f is a real function. a) Prove that if $L = \lim_{x\to a} f(x)$ exists, then $|f(x)|\to |L|$ as $x \to a$ . Proof: Suppose that f is a real function. And suppose $L = lim_{x\to a} ...
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0answers
30 views

Convergence of a monotonically increasing sequence

Given $a_n$ monotonically increasing and $a_n>0$. Which of the following converges? (1) $\frac{1}{a_n^2}$ (2) $e^{-a_n}$. I could not see any reason why both (1) and (2) will not converge. If ...
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0answers
13 views

Limit of a quotient of volumes

Let $f: \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ diffeomorphism defined on a compact subset $\Omega$. I want to see why for any $a \in \Omega$: $\lim_{r \to 0} ...
2
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1answer
51 views

Differentiation under integral sign without DCT

Suppose $f: \Omega \times I \subseteq \mathbb{R}^n \to \mathbb{R}$ is differentiable, where $\Omega$ is measurable and $I$ is an open interval. How do you show that if $\frac{\partial f}{\partial t}$ ...
3
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1answer
67 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
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2answers
349 views

Use mathematical induction to prove Σ n,k=1 (1/k(k+1)) = (n/n+1) for all n in Natural numbers?

This is how far I can get: p(n): nΣk=1 (1/k(k+1)) = (n/n+1) p(1): 1Σk=1 (1/(1+1)) = (1/1+1) => 1/2 = 1/2 p(1) is true. Assume that p(k) is true. p(k) = kΣk=1, (1/k(k+1)) = k/k+1 Show ...
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6answers
343 views

If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges

Let $\displaystyle \sum a_n$ be convergent series of positive terms and set $\displaystyle b_n=\sum_{k=n}^{\infty}a_n$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges. I could ...
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1answer
275 views

Graph of continuous function has measure zero by Fubini

In other occasions, people have asked here how to prove that the graph of a continuous function defined on a box has measure zero. The arguments given where normally: 1) Use the fact that the ...
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2answers
50 views

Fixed point in compact metric space

I guys! I try to solve the following small problem. However, I'm not able to prove the second part. In particular, I have some problems in using the compactness hypothesis on $X$ to find proper ...
5
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2answers
106 views

Which integral is larger?

The question: Given $f$ to be a positive, measurable function on $[0,1]$, which is larger, $\displaystyle\int_0^1 f(x)\log f(x)\,dx$ or $\displaystyle\left(\int_0^1f(s)\,ds\right)\left(\int_0^1\log ...
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2answers
75 views

finding intervals on which f is a continuous inverse

I'm having trouble wrapping my head around this problem. I'm given a function f(x) - x + sinx and told to find all the intervals on which f has a continuous inverse. I honestly really have no idea ...
1
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1answer
78 views

Proof about a subset of a metric space

Prove that a subset $A$ of metric subspace $(P, p')$ of metric space $(M, p)$ is open in subspace $(P, p')$, regarded as a metric space in its own right, if and only if there exists an open set $U$ in ...
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2answers
29 views

Time derivative of operator

I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger ...
3
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1answer
104 views

A continuous function on a sphere attains the same value at some pair of antipodal points [closed]

Let $S=\{x\in \mathbb R^n: \|x\|=1\}$ be the unit sphere in $\mathbb R^n$, and let $f: S\to \mathbb R$ be a real-valued continuous function on $S$. Prove that there is a point a belonging to $S$ such ...
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2answers
64 views

Determine whether the function is Lebesgue measurable

Here's the full problem: Let $\mathcal{N} \subset [0,1]$ be a non-measurable set. Determine whether the function $$f(x) = \left\{ \begin{array}{lr} -x & : x \in \mathcal{N}\\ ...
1
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1answer
35 views

continues function statement in real analysis [closed]

I ran into a challenge, i read following sentence in one note. anyone could describe or prove it for me? F is a continues function at point $ x_0 \Leftrightarrow (x_n \to x_0 \Rightarrow ...
0
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2answers
218 views

Does a Sequence Converge to 0 if and only its Reciprocal Sequence Diverges to Infinity?

I was considering yesterday whether or not the question in the title is, in fact, true. I believe that the definition of a convergent sequence is $(\forall \epsilon>0)(\exists ...
2
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1answer
70 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
2
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1answer
41 views

If $\alpha$ and $\beta$ are complex numbers, then $|\alpha-\beta|^p\leq(|\alpha|^p+|\beta|^p)$?

I have no idea how to show this in the case that $0<p<1$. I can show that $|\alpha-\beta|^p\leq 2^{p-1}(|\alpha|^p+|\beta|^p)$ for $p\geq1$ by using the convexity of the function $x^p$, but am ...
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0answers
37 views

Neighborhood near $0$ for a sequence of non-negative measurable functions can be made arbitrarily small in measure.

I am doing some practice exercises from Avner's FoMA (2.4.1 to be precise). And encountered the following: Let $(X,\Sigma, \mu)$ be a finite measure space. Let $\{f_n\}$ be a sequence of a.e. ...
4
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1answer
35 views

An application of law of large numbers

How can one apply a law of large numbers to a Poisson Process in order to deduce the analytic fact that $$\lim_{t\rightarrow\infty} e^{-t}\sum_{n=0}^\infty ...
4
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0answers
103 views

Closed form of arctanlog series

What tools would you recommend me for $$\operatorname{arctan}\left( \frac{1}{1}\right)\log\left(1+\frac{1}{1}\right)+\operatorname{arctan}\left( ...
2
votes
3answers
109 views

How to prove that S = {$(x,y)| y > x^2$} is open?

I don't understand how to do it at all. My professor tried so patiently to explain it to me but I just don't get it. Here is what he did: Choose any point in S, say (a,b). The point has a ...
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0answers
31 views

Proving equivalence between these two definitions of integrability

I have a definition that say that a measurable function $f $ is integrable if there exists a sequence $\{f _n \} $ of integrable simple functions having the following properties. (a) $\int |f _n - f ...
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2answers
81 views

Find all functions $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, f^n(x)=-x$

I got this problem: Prove that the only function $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, \forall x\in\Bbb{R}, f^n(x)=-x$ where $f^n =f\circ ...
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5answers
335 views

Evaluate $\int_0^{\frac{\pi}{2}} \ln(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
3
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1answer
41 views

Geometric interpretation for $\int_{-\infty}^{\infty} f'(x) f(x) dx = 0$, where $f$ has compact support.

Let $f \colon \mathbb{R} \to \mathbb{R}$ be continuously differentiable with compact support. Then it holds that $$\int_{-\infty}^{\infty} f'(x) f(x)\, \mathrm{d}x = \lim_{n \to \infty} \int_{-n}^n ...
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2answers
40 views

Infimum of a set, not in the set.

Let $E$ consist of all the numbers $\frac1n$, where $n=1,2,3,\dots$ Then $\sup E = 1$, which is in $E$ and $\inf E=0$, which is not in $E$. - Rudin(Principles of mathematical Analysis) This ...
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5answers
504 views

Why is $\sin(xy)/y$ continuous?

Me and my mates are crunching this question for a while now. While we know that $\sin(xy)$ is continuous , $1 / y $ as the other part of the function clearly has a continuity gap at $y = 0 $, though ...
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3answers
54 views

Proving that the sequence $\{x^2 (\cos (1/x) - 1)\}$ does not converge to $0.$

I'm trying to prove whether or not $$ \sum_{x = 1}^\infty x^2 \left (\cos\left (\frac{1}{x}\right ) -1 \right ) $$ converges. Based on graphs, I think that the sequence $\{x^2 (\cos (1/x) - 1)\}$ ...
1
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1answer
52 views

Any necessary and sufficient condition for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$?

I am looking for any established result that is necessary and sufficient for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$ for any real sequences $\{a_n\} \text{and} \{b_n\}\ne 0$, where both ...
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0answers
27 views

Show Cantor function is continuous using sequences

To construct the Cantor set, $C$: Define $J_{1,1}:= (1/3,2/3)$. And define $J_{2,1}:= (1/9, 2/9)$ and $J_{2,2}:= (7/9,8/9)$. Continue to get sets $J_{n,k}$ and define: $K_1 = [0,1] \setminus ...
1
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1answer
28 views

Is the set of non-terminating binary decimals in [0,1] lebesgue measurable?

I think it is true, but I don't know where to start...Any hint would be appreciated!
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3answers
130 views

Problem 17 in chapter 3 of Spivak book

If $f(x)=0$ for all $x$, then f satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(xy)=f(x)f(y)$ for all $x$ and $y$. Now suppose that $f$ satisfies these two properties, but that $f(x)$ is ...
2
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0answers
51 views

$-p_{xx}+f(p)=0$ has a unique solution $p$

In a paper I am reading they have the following as a Lemma without proof. So I am trying to prove it myself. Suppose that $f:\mathbb{R}\to \mathbb{R}$ that satisfies the following: $f'(0) >0$ ...
1
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1answer
32 views

Evaluate $\lim_{h\to 0^+} \frac{e^{-\frac{1}{h}}}{h}$

I am looking for some hints to show that $\lim_{h\to 0^+} \frac{e^{-\frac{1}{h}}}{h} = 0$. I have tried rewriting \begin{aligned} \lim_{h\to 0^+} \frac{e^{-\frac{1}{h}}}{h} &= \lim_{h\to 0^+} ...
0
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1answer
63 views

Advanced Calculus Question. Prove (sn + tn) is a Cauchy sequence

Based on the definition of a Cauchy sequence, that if (sn) is a Cauchy sequence and (tn) is a Cauchy sequence, then (sn + tn) is a Cauchy sequence I try to work from the definition that |(Sn + tn) ...
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0answers
95 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
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1answer
74 views

Uniqueness of non terminating binary decimals in $[0,1]$

How do we prove that non terminating binary decimals has a unique binary expansion? I notice that this question has been posted before but no answer has been given yet...Thanks for your explanation!
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1answer
60 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
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4answers
68 views

Proving $\lim\limits_{n \to \infty}[\sqrt{n^2 + 1} - n] = 0$

I'm trying to prove $\lim\limits_{n \to \infty}[\sqrt{n^2 + 1} - n] = 0$. Is the following a correct proof? For all $n$ we have $0 \leq \left|\sqrt{n^2 + 1} - n\right| \leq \left|\sqrt{n^2+1} - 1 ...
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0answers
38 views

Property of function with given negative variation

Consider a function $f:[a,b]\rightarrow R$, where $-\infty<a<b<+\infty$. Suppose also that $f(x)+x$ is non-decreasing, with $f(b)-f(a)>b-a$. Show that there necessarily exist values ...
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0answers
26 views

Two sequences of functions that are the same almost everywhere, converge to the same limit.

Suppose we have a sequence of simple functions $\{\phi_k\}_{k=1}^{\infty}$ defined on a finite measurable set $E$, that converges point wise to some measurable function $f$ defined on the same set. ...
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0answers
43 views

Which $L \subset [0,1]$ equal the set of limits of a sequence of a sequence in $[0,1] \setminus L$?

I was glanced at this question here and it cause me to wonder the following: Question: Is there a simple description of the subsets $L \subset [0,1]$ with the property that there exists a sequence ...
0
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1answer
35 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
1
vote
1answer
128 views

Why can't solutions to Autonomous ODE intersect?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n $ be continuously differentiable. Why can't solutions to $x'=f(x) $ intersect? I use a proof by contradiction: Assume solutions can, and do intersect ...
0
votes
1answer
40 views

Limit of a quotient. Proof through the definition of absolute and relative errors.

I have difficulty with understanding V. Zorich's (Mathematical Analysis, p.85 of English edition) proof of a $$\lim\limits_{n\to\infty}\frac{x_n}{y_n}=\frac{A}{B}, \quad y_n\neq0,\quad B\neq0$$ Proof: ...
10
votes
1answer
280 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
0
votes
1answer
50 views

If a vector field with zero divergence vanishes on a flat portion of boundary, its normal derivative is zero

Let's consider a vector field $v$ in a bounded region $R$ of the space; assume that $\operatorname{div}v=0$ and $v=0\,\,\text{on}\,\, \partial R$; I have to prove that $$(\nabla v)^Tn=0$$ where $n$ is ...