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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Proof of Bolzano-Weierstrass for functions over countable domains

Theorem A bounded sequence of functions defined over a countable domain has a convergent subsequence. Attempted Proof: Let $f_n$ denote a sequence of functions, where each function is defined ...
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7 views

A Challenge on linear functional and bounding property

i take a midterm exam and after that i wrote it on a paper, my instructor was unable to solve it. i take a picture and insert it here in order to anyone help me. Thanks to evreyone
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1answer
18 views

If $\int_0^{x} g \leq \int_0^x f$ and $\phi$ is nonincreasing then $\int_0^{\infty} \phi g \leq \int_0^x \phi f$

Let $f, g$ be measurable real-valued functions on $[0, \infty)$, with $$\int_0^{x} g \leq \int_0^x f$$ for each $x$. Show that if $\phi: [0, \infty) \rightarrow \mathbb{R}$ is nonincreasing, then ...
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17 views

If $\int_0^1 f(y)\sin(xy) dy = 0$ for every $x$, then $f = 0$ almost everywhere.

Can someone please give me a hint on this question, I have no idea where to start. Let $f \in L^p$ for some $1 \leq \infty$. Assume for all $x \in [0,1]$ that $$\int_0^1 f(y)\sin(xy) dy = 0$$ Show ...
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10 views

Can we find a natural number $m$ such that $[A^{c^{t}}]=[A^{c^{m}}]$?

Let $t$ be a non natural namber. Can we find a natural number $m≠t$ such that $$[A^{c^{t}}]=[A^{c^{m}}]$$ where $[x]$ is the integer part of $x$ (the floor function)? Here $A>1$ nad $c>2$ are ...
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1answer
14 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: ...
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10 views

Find an Upper bound of absolute value (triangle equality application)

Given the functions f(x) and g(x), how can I find a bound for the absolute value \begin{equation} \|f(x)-g(x)-2\| \end{equation} is it correct to say $\|f(x)-g(x)-2\|\leq ...
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14 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim diam E_n $= $0$, prove ...
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1answer
26 views

An element $u$ is an upper bound of $E$ if and only if $t>u$ implies $t\notin E$

Let $S$ be an ordered field and $S \supset E\neq \varnothing$. Then, the following are equivalent: $u \in S$ is an upper bound of $E$. $t \in S$ and $t > u$ implies $t \notin E $. My Try: ...
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19 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
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2answers
26 views

Compute two-dimensional integral

How to compute $\int\int_Mydxdy$, where $M = ${$y \ge0,x^2+y^2\le1,(x-1)^2+y^2\ge1$}? My guess was that $-1 \le x \le 1, \sqrt{1-x^2}\le y\le \sqrt{1-(x-1)^2}$, then integrate $ydxdy$ using these ...
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1answer
45 views

Given $x,y\in\mathbb R$ is there a “formulaic” way to obtain a $q\in\mathbb Q$ with $a<q<b?$

Is there an assignment of reals $x,y$ to a rational number $q(x,y)$ for which $$\forall_{\mathbb R} x.\forall_{\mathbb R}(x<y).\left(x<q(x,y)<y\right)\hspace{.2cm}?$$ For computable reals, ...
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4answers
50 views

If $\dfrac{x_{n+1}}{x_n}$ converges to $l$, then $x_n$ converges to $0$

Suppose that $(x_n)$ is a sequence in $\Bbb{R}$ and that $$\lim_{n\to\infty}\dfrac{x_{n+1}}{x_n}=l$$ for some $l\in(-1,1)$. How do I show that $x_n\to 0$? For any $\epsilon>0$ we have an $N$ such ...
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1answer
52 views

Does $\sum\limits_{k=1}^\infty |\sin(ak)/k|$ converge?

Does $\sum_{k=1}^\infty |\sin(ak)/k|$ converge for all $0<a<\pi$? I do not think so since for $a=\pi /2$: $$\sum_{k=1}^\infty\left\vert\frac{\sin(ak)}{k}\right\vert=\sum_{k=0}^\infty ...
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2answers
36 views

Discontinuous everywhere but range is an interval

Does there exist a function which is discontinuous everywhere but range set is an interval.
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1answer
12 views

Another characterization of the supremum of a set

$u$ is an upper bound of a set $E \subset S$ if given any $\epsilon >0$, there is $\delta \in E $ such that $u - \epsilon < \delta$. PROBLEM: An upper bound $u$ of $E \subset S$ ($E \neq ...
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3answers
51 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
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4answers
34 views

proof that $a_n$ is a null sequence

I want to prove, that $a_n$ is a null sequence if $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1$$ That means that $\forall \epsilon > 0: \exists N \in \mathbb{N}: n \ge N: ...
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0answers
32 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that ...
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21 views

Prove that Recurvisv limits are equal [on hold]

Prove that: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} a_{n-1}$ Ideas?!?
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1answer
33 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
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2answers
20 views

Range of Monotonic function as countable union of intervals [on hold]

Prove that Range of Monotonic function can be written as countable union of disjoint intervals(degenerate or non-degenerate). Please note: I am not asking to prove that a monotonic function are ...
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0answers
16 views

Fourier Series equal to function

If a function is continuous on $\Bbb R$, is periodic with period $2\pi$ and is in fact $C^1$, is that enough for the function to be equal to its Fourier series? My textbook only shows when the ...
3
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1answer
39 views

Is the limit function $f$ continuous if $f_n(x_n)\to f(x)$? [duplicate]

Let $I\subset\mathbb{R}$ be an interval and let $(f_n)$ be a sequence of continuous real-valued functions on $I$. Consider the following statements: $f_n\to f$ uniformly; For every sequence $(x_n)$ ...
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1answer
22 views

If $f \in L^{1}(\Omega)$, $g \in L^{2}(\Omega)$, with $\Omega$ a bounded domain in $\mathbb{R}^n$, then can $f.g$ be in $L^1{\Omega}$?.

I have come up with an argument which is as follows. Please correct me if it makes no sense. Consider $\int_{\Omega}|f.g|dm$. Then if $|f|\in L^1$ then $\sqrt{|f|}\in L^2$. Hence we have ...
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specification of model or formation to maintaine hypothesis [on hold]

Develope a model using a topic of ur choice with one independent variable and four dependent variable( a) explaine Abortion between expectations (b) generate data set of the variable
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43 views

What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
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23 views

Uniform distribution in a cube

I came across the following problem and got stuck. Problem: Let $X_1,X_2,...$ be independent Unif$(-1,1)$ and $S_n=X_1^2+...+X_n^2$. Let $$A_n=\{x\in ...
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2answers
43 views

Suppose that a function $f\colon[0,1]\rightarrow \mathbb{R}$ is continuous, $f\left(0\right)=0$

Suppose that a function $f\colon[0,1]\rightarrow \mathbb{R}$ is continuous, $f\left(0\right)=0$ and $f\left(1\right)>0$. Prove that there is a number $x_0$ where $0\leq x_0<1$ such that ...
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0answers
15 views

Level sets volume

Suppose that $f:\mathbb{R}^d\to\mathbb{R}$ is a nice function (whatever nice should mean), non-negative, with a compact support. Fix $v >0$ and define $$ A_{\epsilon} := \{x\in \mathbb{R}^d : v \le ...
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2answers
26 views

Checking continuity looking whether image set is interval or not

Let $A(\neq \phi)\subseteq\mathbb{R}$. Suppose $f : A \to \mathbb{R}$ is a monotone function such that the image $f (A)$ is an interval. Then prove that $f$ is a continuous function. And if $f(A)$ is ...
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A basic analysis/O.D.E/perturbation theory question

Consider a system of equations $$x'=f(x,y,\epsilon)$$ $$y'=\epsilon g(x,y,\epsilon)$$ I have seen in the book to claim the following: As $\epsilon -> 0$ the limit is $$x'=f(x,y,0)$$ $$y'=0$$ I ...
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1answer
59 views

Assume that $f$ is $2\pi$ continuous and $C^1$ such that $\int_{-\pi}^{\pi} f(x) dx=0$.

Show that $\int_{-\pi}^{\pi} (f(x))^2 dx \leq \int_{-\pi}^{\pi} (f'(x))^2 dx$. So here's my approach to this question: Assume that $f$ was $2\pi$ continuous and $C^1$. Therefore, we have that ...
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1answer
20 views

Finding connected componets of a set of continuous functions

In the metric space (C[0,1], d∞) consider the set: U= {f in C[0,1]: f(x)≠0 for all x in [0,1]} Prove that U is open and find its connected components. Proving that U is open is easy, but I don't ...
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2answers
23 views

Pointwise convergence and Sobolev bounded

If $\{f_n\}$ is a sequence of function on $\Omega$, and $\lim_{n\to\infty}f_n=f$ on $\Omega$. $\|f_n\|_{W^{k,p}(\Omega)}$ is bounded (or uniformly bounded), then whether $f$ is in $W^{k,p}(\Omega)$. ...
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1answer
13 views

Prove that $\nu$ is an outer measure

Let $\eta: P(\mathbb{R}) \to [0,\infty]$ be an arbitrary function with $\eta(\emptyset)=0$, where $P(\mathbb{R})=\{A: A \subset (-\infty,\infty)\}$. For $A \subset \mathbb{R}$ (i.e. $A \subset ...
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1answer
17 views

Cauchy Product $n$ times

I'm looking for a short and precise proof of the following identity; $$\left(\sum_{k=0}^\infty \frac{C_k}{k!}x^k\right)^n=\sum_{k=0}^\infty\left[ ...
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1answer
32 views

Verifing that $f$ is integrable

$f:[0,1]\times[0,1]\to \Bbb R$ be given by $$f(x,y) = \begin{cases} xy^2, & \ y\lt x^2 \\ x+2y, & \ y\ge x^2 \end{cases}$$ I need to show that $f$ is integrable. My idea is that to show ...
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decide if the set function $\nu$ defined is a measure

Let $(X,F,\mu)$ be a measure space and let $A \in F$ with $\mu(A)>0$. In each case below, decide if the set function $\nu$ defined there is a measure. Prove that it is a measure or provide a ...
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1answer
30 views

example for Lebesgue measure [duplicate]

Construct a closed, uncountable, perfect, nowhere dense subset of $[0,1]$ which has Lebesgue measure $\frac{1}{2}$. (Hint: Find the Cantor subset of $[0, 1]$ with Lebesgue measure $\frac{1}{2}$)
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4answers
106 views

Showing that $\lim_{x\to\infty}\left(\sqrt{x^2+c}-x\right)=0$

A limit I had to compute recently boiled down to the following limit: $$\lim_{x\to\infty}\left(\sqrt{x^2+c}-x\right)=0\quad\mbox{for $c\ge0$}$$ How can I show that this limit is correct?
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23 views

uniform boundedness of a class of functions

Consider a class of (proper closed) convex function on $[0,1]^d$, which we shall denote $\mathcal{F}$. If every element of $\mathcal{F}$ is bounded in $L_2$, say $$\int_{[0,1]^d} |f(x)|^2\ dx\leq 1,$$ ...
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1answer
11 views

Existence of Disks around points in Sequentially Compact Space

Theorem: There is an $r>0$ such that for each $y \in A$, where $A$ is a sequentially compact metric space, $D(y,r) \subset U_i$ for some $U_i$. Proof: Suppose not. Then for every integer $n$, ...
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40 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
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1answer
18 views

Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset ...
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17 views

Choosing a clever “test function” in Sobolev spaces.

Given $\mathbf{f}$ with $f_1,...,f_N\in L^2(\Omega)$ $$\int_\Omega \mathbf{f} \cdot \nabla v = 0 \quad\forall v \in H_0^1(\Omega)$$ we have $\mathbf{f} = \mathbf{0}$ a.e. since $\mathbf{f} \in ...
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2answers
34 views

Suppose that the function $f:[0,1]\rightarrow \mathbb{R}$ is continuous and that $f\left(x\right)>2$

Suppose that the function $f:[0,1]\rightarrow \mathbb{R}$ is continuous and that $f\left(x\right)>2$ if $0\leq x<1$. Is it necessarily true that $f\left(1\right)>2$? My attempt: Yes, using ...
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2answers
12 views

Convergent Subsequence and metric

Consider a sequence $y_n$ in a metric space. If $d(y_n, y_m) \geq \epsilon$ for all $n$ and $m$, why does this imply that $y_n$ has no convergent subsequence? My understanding is that this implies ...
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0answers
16 views

Weak convergence in $l^p$-space [on hold]

Let $1<p<\infty$, $x_n=(x_n^{(j)})_j\in l^p$ for $n\in \mathbb N$ and $x=(x^{(j)})_j\in l^p$. Show that $$x_n\rightharpoonup x\iff \forall j\in \mathbb N:x_n^{(j)}\rightarrow x^{(j)}, ...
3
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1answer
72 views

Taylor series of a definite integral

Consider the function of a parameter $\alpha > 0$, given by $$f(\alpha) = \frac{2}{\sqrt 2\pi} \int_0^\infty e^{\dfrac{-x^2}{2\alpha^2}}\cosh(x)\log\cosh(x) dx.$$ From Wolfram-alpha, it seems that ...