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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42 views

Given two real functions, $f$ and $g$, if $|f(x)|<1$ then $|g(f(x))|<g(1)$? Why?

It seems trivial for a certain $x$ but can we say it for all x?
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1answer
11 views

Showing that space of real convergent sequences is complete

Let C be the space of real convergent sequences with $l^{\infty}$ norm. I'm trying to show that it's complete. Let $(x^i_n)_n$ be Cauchy. Then given $\varepsilon > 0$, we have $sup_i |x^i_n - ...
0
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2answers
43 views

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

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)=0$$ converges if and only if $\sum_{n=1}^\infty a_n=\infty$. Proof of $\Rightarrow$: Assume that ...
0
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0answers
8 views

Does locally Lipschitz in one variable imply uniformly Lipschitz in another?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $f(t,x):D ...
0
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1answer
19 views

Construction of sequence from convergent susbsequences

Is it possible to construct the following? A sequence that contains subsequences converging to every point in the infinite set $\{{1, 1/2, 1/3, 1/4, 1/5, ...}\}$ and no subsequences converging to ...
0
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1answer
15 views

Distance between two points with an stimated time.

I am working on a project and one problem is this: I have two points, $A$ and $B$, but the only thing I Know is, time. with a time I want a distance between $2$ points, I want to calculate meters... ...
0
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0answers
10 views

prove that a function is differentiable

let $N \in \mathbb{N}$ be given and consider the mapping $\textrm{pow}_N:\textrm{Lin}(\mathbb{R^n} ,\mathbb{R^n})\rightarrow \textrm{Lin}(\mathbb{R^n} ,\mathbb{R^n}) $ defined by ...
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1answer
18 views

An example of the set of distances of two points in two different closed sets having no infimum

On a problem set for my Analysis in Several Dimensions class (basically real analysis on multivariable functions), I encountered this question: Let $(X, d)$ be a metric space, let $C ⊂ X$ be a ...
0
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0answers
17 views

Infinite Cartesian Product of FINITE SET is Contable?

We Know that infinte cartesian product of Countable set is Uncountable... but Countable means Either Finite or Countably Infinite... If i consider that countable set is finite, then the Catesian ...
0
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1answer
14 views

$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$

Le $f,g : \mathbb{R} → \mathbb{R}$ be extended integrable functions. Let $λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$. Give necessary and sufficient conditions on $f,g$ for $λ ⊥ ρ$ and necessary ...
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0answers
19 views

How to calculate these improper integrals?

Let $0<\alpha<1$. $B_R$ denotes open ball in $\mathbb{R}^N$($N\geqslant 2$) with center at $0$ and radius R. How to compute the following integrals: $$ \int_{\mathbb{R}^N\setminus ...
1
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1answer
29 views

Prove $ f(c)\int_{a}^{b}g(x)dx=\int_{a}^{b}g(x)f(x)dx$

Assume that $f:[a,b]\rightarrow\mathbb{R}$ is continuous on $[a,b]$ and $g:[a,b]\rightarrow\mathbb{R}$ is integrable and $g(x)\geq0$ for all $x\in[a,b]$. Then there exists a $c\in(a,b)$ such that ...
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0answers
2 views

The $\mu^{*}$ measurable set of Riesz–Markov–Kakutani representation theorem

In the proof of Riesz–Markov–Kakutani representation theorem, we define $\mu^{*}(V)=\mbox{inf}\{\mu(U),V\subset U\}$ where $U$ is open, it is quite obvious that such definition gives an outer measure, ...
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0answers
4 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
0
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2answers
36 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 ...
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0answers
32 views

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

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
2
votes
2answers
37 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 ...
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2answers
54 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
38 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 $a>0$, ...
1
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1answer
11 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
25 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
vote
1answer
70 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 ...
1
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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 ...
0
votes
2answers
34 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
votes
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 ...
0
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1answer
21 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 $\implies$ ...
0
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3answers
58 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?
0
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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 ...
0
votes
1answer
12 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 ...
0
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1answer
48 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 ...
-2
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2answers
67 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 ...
1
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3answers
62 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 ...
0
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0answers
30 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
votes
2answers
79 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 ...
0
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1answer
16 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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votes
1answer
21 views

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

i am not able to solve this please if somebody could help
1
vote
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 ...
1
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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 ...
0
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0answers
5 views

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

Consider "random strings" over an $m$-letter alphabet where we are looking for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable ...
0
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1answer
31 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
1
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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 ...
1
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1answer
28 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 ...
1
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0answers
15 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.
-1
votes
0answers
30 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
votes
1answer
33 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 $M$. ...
0
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0answers
25 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
votes
1answer
80 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
votes
1answer
33 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!
1
vote
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
37 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