For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag. Do not use this tag for questions on solving equations.

learn more… | top users | synonyms

0
votes
1answer
28 views

Show that the closure operation has the following properties

Let $E_1$ and $E_2$ be subsets of $\mathbb{R}$. I need to show that the closure operation has the following properties: a)$(E_1 \subset E_2) \Rightarrow (closure(E_1) \subset closure(E_2))$ b)$...
4
votes
2answers
72 views

What properties of the real numbers are almost always true and there are no (or very few) known examples of?

What properties of the positive real numbers are almost always true and there are no (or very few) known examples of? Two that come to mind are numbers that are normal in every base and numbers ...
3
votes
5answers
81 views

How are sets “detached” from their structure?

This question is best asked with an example. Consider the real numbers. However we construct the real numbers, the "final product" so to speak, is not just a set, but it is a complete ordered field. ...
0
votes
1answer
27 views

Find the Limit point of this exercises

Good morning, i'm working in this exercise and i solve this, but, i don't know it's fine, please how you can find the limit point? 1) $\left\{ 1-\frac{1}{n}\::\:n=1,2,3...\right\}$ Well, i say the ...
2
votes
1answer
87 views

Value of the integral $\int_{0}^{1}x^{n}(1-x)^{n}dx$

If $A=\int_{0}^{1}x^{n}(1-x)^{n}dx$ then which of the following is/are true? $1.$ $A$ is not a rational number. $2. 0<A\leq 4^{-n}.$ $3.$ A is a natural number. $4.$ $A^{-1}$ is a natural ...
2
votes
1answer
28 views

Confusion in finding left and right hand limits [duplicate]

Let $f:\mathbb R$→$\mathbb R$ defined as - $f(x)=0$, if $x$ is irrational or $x=0$ and $f(x)=1/q$, if $x=p/q$, $p\in$$\mathbb Z$ ,$q\in$$\mathbb N$, $(p,q)=1$. What are the points of continuity of $...
0
votes
1answer
46 views

Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
5
votes
1answer
51 views

For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
7
votes
1answer
87 views

Is $\pi = 3.14159…$ first-order definable in the reals?

Given first-order logic with equality and the real field $\mathbb{R} = (R, 0, 1, <, +, \cdot)$, is $\pi$ first-order definable? By first-order definable, I mean a sentence of the form $\exists x \;...
3
votes
6answers
684 views

Real and Non-real Numbers; Value of Zero? [closed]

There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-...
-4
votes
0answers
50 views

Irrationality of a number [closed]

Is there any proof for $\sqrt3$ being an irrational number where we are not forced to conclude that $\sqrt3$ is irrational number?
3
votes
4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
0
votes
2answers
88 views

Why we square while doing the proof of √2 is irrational? [closed]

When we prove that $\sqrt 2$ is irrational by the method of contradiction, we assume $\sqrt 2$ is a rational number: $\sqrt 2 = a/b$ Squaring both sides, $2 = a^2/b^2$. Here is my question: is ...
1
vote
2answers
70 views

$\aleph_1$ and $\omega_1$, what are they?

Sorry for my ignorant question but.. I understand that some sources says that $\aleph_1$ is the cardinality of the real numbers (ℝ) because In set theory $$\mathfrak{c} = 2^{\aleph_0} $$ and the ...
0
votes
0answers
96 views

An equivalence relation on Dedekind cuts

The definition of a cut $(A,B)$ is: $$ \mathbb{Q} = A \cup B \\ A \ne \emptyset ,B \ne \emptyset \\ \forall a \in A,\forall b \in B,a < b \\ $$ Define a relation on the set of all cuts of $...
0
votes
2answers
40 views

Listing real numbers as countable like listing rational numbers [closed]

like proving the set of positive rational numbers are countable, where we list the rationals as the following list, why can't we represent real numbers like the same? If positive Rational numbers (p/...
4
votes
4answers
213 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
1
vote
1answer
77 views

approximate irrational numbers by rational numbers

I want to prove this below: (1) For any irrational number $\alpha$, there exist infinitely many rational numbers $\frac{m}{n}$ such that $\left| {\alpha - \frac{m}{n}} \right| < \frac{1}{{{n^2}}}$...
0
votes
1answer
34 views

A bounded interval covered by finite open intervals

If a bounded closed interval $[a,b]$ is covered by finite open intervals $\bigcup\limits_{j = 1}^n {({c_j},{d_j})} $, I want to prove $b - a < \sum\limits_{j = 1}^n {({d_j} - {c_j})} $. It seems ...
0
votes
0answers
20 views

Poll Ranking Formula

I am having a hard time making a ranking formula for a project I am working on. Here's a practical approach and I would really appreciate any help. Lets say we have $1000 to share among 5 people,...
0
votes
1answer
38 views

Countable Set, the numbers rational

Good morning, i want to try solving this exercise: Prove $\mathbb{Q}$ is countable set. I make this: $f:\mathbb{N}\rightarrow\mathbb{Q}$ Be q $∈$ $\mathbb{Q}$, if $q>0$ then: $f\left(n\right)=\...
15
votes
3answers
875 views

Does the concept of permutation make sense for a set indexed by the real numbers?

I know that the concept of permutation makes sense for sequences, which are sets indexed by the natural numbers (if the sequence is infinite) or indexed by the first $n$ natural numbers (if the ...
2
votes
5answers
258 views

Why does an argument similiar to $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+…=1$ show that $2+4+8+…=-2$ [duplicate]

See how to prove $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=1$ $x=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ $2x=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ Then: $x=1$ Now I use the same argument to ...
2
votes
1answer
107 views

How many points can we remove from $\mathbb{Q}$ so it is still dense in $\mathbb{R}$?

I want to check whether: $\mathbb{Q}\backslash\{0\}$ is dense (or not) in $\mathbb{R}$ $\mathbb{Q}\backslash F$, $F$ is a finite set in $\mathbb{Q}$, is dense (or not) in$\mathbb{R}$ $\...
0
votes
2answers
51 views

What is $\mathbb{R}^+$

Well some books refer to $\mathbb{R}^+$ to be the set of all positive real numbers while others say $\mathbb{R}^+$ is a set of non-negative real number. Is there a universally accepted definition ...
7
votes
2answers
146 views

Is there a mathematical statement that is linking integer limits to real limits?

I saw a question asking for the limit $$\lim_{n \to \infty}\frac{\tan(n)}{n}.$$ At first I thought that the limit assumed $n$ to be a real number. So I gave the advice to use $\pi/2+2\pi k$ and $2\...
0
votes
3answers
27 views

show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing

How do I show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing? I know that $\frac{1}{n}$ is monotonically decreasing, but I am unsure how to show $f_n$ is monotonically decreasing ...
1
vote
2answers
62 views

How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
10
votes
3answers
221 views

Solve the equation $2^{1-x} + 2^{\sqrt{2x-x^2}}=3 $

Solve the equation $$2^{1-x} + 2^{\sqrt{2x-x^2}}=3 \tag 1$$ on reals, using elementary knowledge (using trigonometry or logarithms is allowed, but without limits, differential calculus etc.) We ...
0
votes
0answers
54 views

$f \in L^1(\mathbb{R})$ implies there exists $a \in \mathbb{R}$ with $\int_{(-\infty, a]} f = \int_{[a, \infty)} f$

In studying for a qualifying exam, I found a problem asking me to prove: If $f \in L^1(\mathbb{R})$ has $\int_\mathbb{R} f \neq 0$, then there exists $a \in \mathbb{R}$ with $\int_{(-\infty, a]} f ...
2
votes
1answer
81 views

Well-orderings of $\mathbb R$ without Choice

The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
0
votes
1answer
6 views

finite partitions of the square that separate all equipotent sets of points

This question asked whether there exists a finite partition of $[0, 1]^2$ and a finite set of points in $[0, 1]^2$ that can't be affinely transformed to fall into one part of the partition. I would ...
0
votes
1answer
15 views

Question about supreme of infinity and supreme of empty set

Good morning, i have a few question about supreme. 1- ¿What happen with the supreme of infinity? I think not have a supreme, because you need a upper bound, but i dont know. 2- ¿What happen with ...
1
vote
1answer
43 views

In what space is a closed set is not or not necessarily $G_\delta$

We know that A closed set in a metric space is $G_\delta$ Is there any topological space where a closed set is not necessarily $G_\delta$? I am thinking a space where singletons are well known to be ...
1
vote
1answer
29 views

Cardinality of infinite words from finite alphabet is the same as that of $\mathbb{R}$

Let $[n]$ denote the set $\{0, 1,2, \ldots, n\}$. I want to show that $[n]^{\mathbb{N}} \simeq \mathbb{R}$. I am aware you could do this with base $n$-ary expansions of reals, but that seems a bit ...
3
votes
0answers
44 views

$f(f(…f(x)…))$ $a$ times, where $a\in\mathbb{R}$

Take $f(x)$ and do a "double-call": $f^2(x)=f(f(x))$ I use this notation here to explain my problem. This can be easy calculated for any function. Also $f^{100}(x)$ is not really a problem. This ...
18
votes
6answers
2k views

when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a ...
2
votes
4answers
39 views

Denseness of set S in R [duplicate]

Consider the set $S= \{\frac{p}{2^n} : p,n \in \mathbb{Z}\}$. Is it dense in $\mathbb{R}$ ?? To me intuitively it seems to be dense but I cannot come up with any analytic proof or disprove .
1
vote
3answers
45 views

Are there locally compact subsets of $\mathbb{R}$ which are not compact?

Are there locally compact subsets of $\mathbb{R}^n$ tht are not compact? A set $X \subset \mathbb{R}^n$ is compact if it is closed and bounded. Every infinite set $\{ x_n \}$ has a subsequence $\{...
1
vote
2answers
70 views

For any consecutive natural numbers $a_1,a_2$ are there infinitely many primes $p,q$ such that: $a_1<\dfrac{p}{q}<a_2$?

Progress: Let $a_1,a_2$ consecutive natural numbers; prove or disprove the infinitude of distinct prime pair $p,q$ which satidfies: $a_1<\dfrac{p}{q}<a_2$ The most challenging part of the ...
0
votes
2answers
26 views

Prove that $M_1^2\leq 2M_0M_2$, if $2M_1t≤2M_0+M_2t^2$

Let $0\leq M_1,M_2,M_3\in\mathbb{R}$ and $\forall \ t\in\mathbb{R}:\ 2M_1t≤2M_0+M_2t^2$. Prove that $M_1^2\leq 2M_0M_2$. I tried assigning different values to $t$, but this didn't help.
0
votes
1answer
27 views

in which subset of $R^2$ the series is convergent? [duplicate]

For $(x,y) \in \Bbb R^2 $ ,consider the series $\lim_{n \to \infty } \sum_{l,k=o}^n \frac{k^2x^ky^l}{l !} $ .Then the series converges for $ (x,y)$ in 1.$(-1,1)\times (0, \infty )$ ...
1
vote
1answer
75 views

Find the minimum value of expression involving real numbers

Let $n$ positive integer. Find the minimum value of expression: $$ E=max(\frac {x_1} {1+x_1},\frac {x_2} {1+x_1+x_2}, ... , \frac {x_n} {1+x_1+..+x_n})$$ where $x_1,x_2, .. , x_n$ are real ...
1
vote
0answers
22 views

Negative of arbitrary lower bound is arbitrary upper bound?

Statement: Let $A \subset \mathbb{R}$ be bounded above. Prove that $\inf\{x\in\mathbb{R}: -x\in A\} = -\sup(A)$. Proof: Since $A$ is (implicitly nonempty and) bounded above, $\sup(A)$ exists. Denote $...
0
votes
0answers
58 views

Rigorous Approach to Precalculus

I've made the mistake of looking at more advanced texts that deal with precalculus-level mathematics in a more formal, rigorous way than usual. Perhaps this isn't a mistake, but now that I've glimpsed ...
2
votes
3answers
50 views

What makes negative numbers different from positive numbers other than their being (almost) opposite?

To quote from Wikipedia's article on negative numbers Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive ...
1
vote
1answer
32 views

Finding Limit Points?

I have two sets: $ A = ({{ (-1)^n + 2/n : n = 1, 2, 3, ...}}) $ and $ B = ( x \in \mathbb{Q} : 0 < x < 1 ) $ How does one go about finding the limit points for these sets? Would I just do $ ...
1
vote
2answers
27 views

Density of real numbers and density function

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the ...
0
votes
1answer
20 views

Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.

As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations. ...
-1
votes
1answer
41 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...