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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11 views

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$.

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)$$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$. I know this question is similar to one I just ...
-4
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0answers
26 views

Show that $\frac {f(x)} x$ is differentiable on $(a,b)$

Show that $h(x) = \frac {f(x)}{x} $ is differentiable, where f is continuous, differentiable and defined on $(a,b)$, where a and b are strictly positive
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2answers
33 views

Is $2^{\infty}$ an Indeterminate form

We know that when $$\lim_{x \to a}f(x) \to 1$$ and $$\lim_{x \to a}g(x) \to \infty$$ then $$\lim_{x\to a}f(x)^{g(x)}$$ is an indeterminate form since in the neighbourhood of $a$ we cannot predict the ...
0
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2answers
38 views

(exercise from Tao's analysis book) Proof of a lemma relating to power set of X

I'm stuck at one exercise from chapter of sets from Terence Tao's analysis book. I need to proof the lemma: Lemma: Let $X$ be a set. Then the set $\{Y : Y \:\text{is a subset of}\: X\}$ is a set. ...
1
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2answers
29 views

Suppose $a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1+a_n)$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$.

Suppose $a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1+a_n)$$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$. Hint: Use the fact that for any $a>0$, ...
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1answer
10 views

Question about Folland's proof of extension-of-premeasures theorem

Here is an excerpt from Folland's Real Analysis. I don't understand why the calculation $\nu (E)\leq \sum _n \nu (A_n)=\sum _n \mu_0(A_n)$ implies $\nu(E)\leq \mu (E)$. Why is this? The $A_n$ are not ...
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0answers
15 views

Approximating $\prod_{r=s}^t (1-b/r)$

I am currently trying to place an order of precision on the approximation $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) \approx \left(\frac{s}{t}\right)^b$$ This follows because $$\prod_{r=s}^t ...
1
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1answer
63 views

Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?

$f$ is injective at an interval $[a,b]$, but discontinuous at some point $c$ in the same interval. I need to prove that its inverse is continuous at that interval. Should I consider what is the ...
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2answers
29 views

Bounds on function $\exp(-\frac{1}{2}x^2)$

I have the following function : $$f(x)=\exp(-\frac{1}{2}x^2),$$ where $x >0$. I am looking for some tight bounds (upper bound and lower bounds) on $f(x)$. Any idea ? P.S.: The problem arises when ...
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2answers
31 views

Let f be continuous. By EVT there exists a c such that f(c)=supx f(x). Show that f is not injective.

I am given a continuous function f in an interval [a,b]. To show that f is not injective, should I consider the definition of the extreme value theorem? I am not sure how to show that it is not one ...
0
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1answer
24 views

On a recursive sequence exercise $a_{n+2} = \frac{4 + 3a_n}{3 + 2a_n}$.

As part of an exercise I am given a sequence defined by $a_1 = 1$ and $$a_{n+1} = 1 + \frac{1}{1 + a_n}$$ I have noticed that the even sequence is decreasing and I want to prove this, the even ...
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1answer
20 views

Cauchy continuous implies standard continuity

Let $f$ be Cauchy continuous. $f$ is Cauchy continuous if for any Cauchy sequence $\{x_{n}\}$ in $(X,d_{X})$, $\{f(x_{n})\}$ is a Cauchy sequence in $(Y,d_{Y})$. Show that Cauchy continuous $\implies$ ...
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3answers
52 views

A function $f$ continuous and injective is monotone.

Question: Let $f: [a,b] \to \mathbb R$ be continuous and injective. Show that $f$ is monotone. Should I consider the contrapositive? Also will considering its derivative help in this example?
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1answer
28 views

Fixed points theorem applications

I want to find the number of solutions to the equation $\cos x=x^2$ in the closed interval $[-\pi/2,\pi/2]$. My approach to the question was to draw the graph of both these functions and see the ...
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0answers
7 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
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0answers
20 views

Regarding Apostol's theory of integration

I have some questions regarding the theory of integration as discussed in Tom Apostol's Calculus. Integration is defined using step functions. My question is, is this definition he presents equivalent ...
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2answers
52 views

An exercise on uncountable subsets of $[0,1]$ [on hold]

I am stuck on how to prove these three questions, or even how to draw the sets so I can see where they overlap. Any help would be appreciated! Let $C\subseteq [0,1]$ be uncountable. Show there ...
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3answers
60 views

Alternative Proof that $\sqrt{p}$ is Irrational when $p$ is Prime

I have found various proofs that $\sqrt{p}$ is irrational on this site, but I didn't find one similar to the one that I am about to post, so I am wondering if it is free of logical problems. Here is ...
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0answers
28 views

Prove the following property of simple functions [on hold]

If $f,g$ are simple functions on $[a,b]$ and $f \leq g$ on $[a,b]$ then $$\int_a^b f \leq \int_a^b g.$$
2
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2answers
78 views

Prove that if $\sum a_n$ converges, then $na_n \to 0$. [duplicate]

Let $a_n$ be a decreasing sequence of nonnegative real numbers. Prove that if $\sum a_n$ converges, then $na_n \to 0$. Hint: use that $n\, a_{2n} \le a_{n+1}+\cdots + a_{2n}$ I couldn't ...
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1answer
12 views

Metric spaces inside of metric spaces

Let $(X, d)$ be a metric space, $Y$ ⊂ $X$ and consider the metric space $(Y, d)$. Show that every open set $U$ in $Y$ has the form $U$ = $V$ ∩ $Y$ for an open set $V$ ⊂ $X$. Show that ...
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1answer
18 views

differential equation with non differentiable non homogeneous part [on hold]

i am not able to solve this please if somebody could help
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1answer
47 views

prove the inequality $0< \frac{1}{m}+\frac{1}{n}+\frac{1}{p}< \frac{47}{60}$

I have an Olympiad Problem, let $m$, $n$ and $p$ denote three natural numbers where: $$m>n>p>2$$ prove that : $$0< \frac{1}{m}+\frac{1}{n}+\frac{1}{p}< \frac{47}{60}$$ I've been ...
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1answer
46 views

To prove or refute: $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1$ then $f \in R\left( \left[ 0, 1 \right] \right)$ [on hold]

Let $f : \left[ 0, 1 \right] \to \mathbb{R}$ such that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1.$$ Then, $f \in R\left( \left[ 0, 1 \right] \right)$ and ...
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0answers
5 views

Getting the shape (or bounding tail estimates) of a probability distribution from its generating function

My original motivation was the question: in the digits of $\pi$, where do we first encounter $10$ consecutive identical digits? (The answer is that there are $10$ consecutive $6$s at position ...
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1answer
30 views

prove properties of upper and lower integrals [on hold]

How to prove these properties of upper integrals? I am trying to use limit rules and/or a sequential approach but cannot figure it out
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1answer
21 views

Existence of a sequence related to the convergence of a series

Trying to prove an exercise, I arrived at the following question: Let $\{a_j\}\subseteq\mathbb{R}^+$ be a monotone increasing sequence with limit $+\infty$. Suppose that there is a $D>0$ such that ...
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1answer
27 views

Show that function has maximum in a given interval

This is a question from my exam in Calculus I: Problem 4 Prove or disprove: [...] a) The function $f(x) = \left(\sin(x) + \sqrt{\log(1+x^2)}\right)^3 e^{\cos(x) - 1}$ has a maximum ...
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0answers
13 views

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. [duplicate]

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. I'm having real trouble proving this inequality. I'd greatly appreciate any help.
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0answers
28 views

Prove that every simple function is Riemann integrable. [on hold]

Prove that every simple function on $[a,b]$ is Riemann-integrable. I understand why this is true, but I am not sure how to go about proving it. Does it have to do with the upper and lower integrals ...
0
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1answer
32 views

A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
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0answers
23 views

Categorically deducding measurability of sections

Two lemmas which are often proved in elementary measure theory courses are that sections of measurable sets are measurable, and sections of measurable functions are measurable. Note $E_x= \left\{y\in ...
12
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1answer
79 views

Function $\Bbb Q\rightarrow\Bbb Q$ with everywhere irrational derivative

As in topic, my question is as follows: Is there a function $f:\Bbb Q\rightarrow\Bbb Q$ such that $f'(q)$ exists and is irrational for all $q\in\Bbb Q$? For the sake of completeness, I define ...
0
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1answer
31 views

What is the name of the following partial differential operator?

What is the name of the following partial differential operator? $$\sum_{|\alpha| \leq n} a_\alpha (\frac{\partial}{\partial x})^\alpha$$ Thank you!
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1answer
26 views

A convergent series of irrational numbers only which is not absolutely convergent

While solving another problem I stumbled upon this. I wonder if such a series exists: "a convergent series of irrational numbers only which is not absolutely convergent". I am thinking but I cannot ...
3
votes
1answer
36 views

To evaluate integral using Beta function - Which substitution should i use?

$$\int_{0}^{1} \frac{x^{m-1}(1-x)^{n-1}}{(a+bx)^{m+n}}dx = \frac{B(m,n)}{(a+b)^ma^n}$$ I have to use some kind of substitution but i do not understand what i use and why ? Thanks
5
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1answer
46 views

Express a real number as a product

Hi guys if I have a number $x \in [1,2)$ is it possible to express such number as: $$x = \prod_{j=0}^{+\infty} (1 + \alpha_j 2^{-j})$$ where each $\alpha_j \in \left\{-1,0,1\right\}$? If yes, how ...
0
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2answers
11 views

Comparing the asymptotic growth of certain functions.

Consider some nonconstant functions $f(x)$ and $g(x)$, and suppose $\lim_ {x \to a} f(x) > \lim_{x \to a} g(x)$ for some nonzero $a$. Can we somehow conclude that $f(x) - g(x)$ is of constant sign ...
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1answer
58 views

To show that $\int_{0}^{\infty} \frac{e^{-ax}\sin(bx)}{x}=\arctan \left( \frac{b}{a}\right)$ [on hold]

To show that $\int_{0}^{\infty} \frac{e^{-ax}\sin(bx)}{x}$ = $\arctan \left( \frac{b}{a}\right)$ I have to use Frunalli integral to do this. But i am not able to start.
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0answers
9 views

Represetation of a smooth function in the neighborhood of its zero-set

Consider $k$ smooth functions $g_i(x)$, $x\in \mathbb{R}^n$, $k<n$. The set $G$ is defined as $G=\{x\in \mathbb{R}^n|g_i(x)=0, i=1,...,k\}$. Let $F(x)$ be a smooth function s.t. $F(x)=0$ for any ...
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2answers
25 views

Different versions of Bolzano Weierstrass Theorem and their relationships.

Which one is the Bolzano Weirerstrass Theorem? Theorem 1. Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. OR Theorem 2. Every sequence of real numbers has a monotonic ...
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0answers
24 views

Discontinuous parametric integral function

Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function ...
1
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1answer
37 views

limit of the following telescoping sequence

The answer i got is $0.25 $.beacuse evey term gets cancelled like telescoping sequence. am i correct?
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2answers
40 views

Is the function $\frac{f(x)}x$ increasing, if $f(x)$ is convex? [on hold]

Suppose $x$ is real and positive. $f''(x) > 0$ (f is convex) is the function $h(x) = \frac{f(x)}x$ increasing? If so, is it strictly increasing? If yes, why? Thank you.
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2answers
61 views

how to show that $\{x\in \mathbb R^n: f(x)=b\}$ is closed

(1) Let $f: \mathbb R^n \to \mathbb R^m$ be a continuous mapping. Let $b\in \mathbb R^m$. Show $$\{x\in \mathbb R^n: f(x)=b\}$$ is a closed set. My thought: I want to show that the set ...
0
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0answers
26 views

Simply connectedness of spherical shell

Consider a spherical shell $U$ in $R^3$(the open region between two spheres). I want to show that any closed curve in $U$ can be shrunk into a single point without leaving $U$. This exercise appears ...
2
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0answers
45 views

How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
2
votes
1answer
49 views

integral involving greatest integer function [on hold]

Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n=\int_1^n \frac{x-[x]}{x^2}dx$. Then, what is $S_{10} + T_{10}$? The only clue that i can get is break the limits of integration according as the ...
1
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0answers
21 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
0
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0answers
30 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?