For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.

learn more… | top users | synonyms

0
votes
1answer
33 views

How to prove that $(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $ for $a,b\in [0,1]$ and $n\in\mathbb{N}$?

Let $a,b\in [0,1]$ and $n\in\mathbb{N}$. Prove the following inequality: $$(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $$ I thought on using M Induction: Assuming that the inequality holds for $n=k,$ ...
6
votes
1answer
85 views

Find the values of the positive integers $n$ such that: $\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$ is positive integer

My question is as follow: Find the values of the positive integers $n$ such that: $$\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$$ is positive integer. I can see that for $n=1$ (among some ...
2
votes
1answer
36 views

Why can't the definition of convergence be alterted to this one?

I am trying to find out of a seqence with the following property is convergent: Let $(r_n)$ be a sequence of real numbers. Suppose there is a number $r\in\mathbb{R}$ such that for any ...
2
votes
1answer
47 views

Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
-4
votes
1answer
63 views

How to describe the Cartesian product $\mathbb{R} × \mathbb{R}$?

I am taking a discrete mathematics course in the spring and in an attempt to fully understand the material I am reading ahead. I came across this statement Let $\mathbb{R}$ denote the set of all real ...
4
votes
4answers
67 views

$\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ dense in $\mathbb{R}$? [duplicate]

I'm guessing $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ is dense in $\mathbb{R}$. I'm having a mental block. How do you show that? (This is motivated by a different hypothesis: if $f$ is ...
3
votes
1answer
43 views

Show that $≺$ is a total ordering

Let $ℕ$ be the set of positive integers. Let $D(n)$ denotes the number of divisors of $n$. We define this binary relation: $n≺m⇔n≤m$ and $D(n)≤D(m)$ where $≤$ is the usual ordering in $ℕ$. Show ...
0
votes
3answers
53 views

Can someone clarify the Archimedean property of the real numbers?

If $x$ and $y$ are real numbers with $x>0$, there exists a natural $n$ such that $nx>y$. Is this basically saying that for any real number, you can find a natural number that is bigger than it? ...
11
votes
4answers
929 views

Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
1
vote
1answer
54 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
1
vote
2answers
22 views

Prove proposition on real numbers and uniqueness.

How would I go about proving the following proposition. Do I have to prove uniqueness, or that if $x^2 = r$, then $x = \sqrt r$? Prove given any $r \in \mathbb R\gt 0$, the number $\sqrt r$ is ...
1
vote
3answers
46 views

Prove if $x ∈ \mathbb{R}$, such that $0 ≤ x ≤ 1$, and $m,n ∈\mathbb{ N}$, with $m ≥ n$. Then $x^m ≤ x^n$

How to prove the following prop. Let $x \in \mathbb{R}$, such that $0 \le x \le 1$, and $m,n \in\mathbb{ N}$, with $m \ge n$. Then $x^m \le x^n$. I don't exactly know where to begin with this proof, ...
-1
votes
1answer
19 views

Prove proposition on real numbers

How would I go about proving the following proposition. Thanks. If $r < 0$ there exists no $x \in\Bbb R$ such that $x^2 = r$.
1
vote
2answers
42 views

About the Erdős-Borwein Constant

The Erdős-Borwein Constant can be found in http://mathworld.wolfram.com/Erdos-BorweinConstant.html My question is : Is there is a document or website containing the value of $E$ with a bigger number ...
0
votes
2answers
42 views

How to argue that one point on real line is on the right of the other?

I have the two points on the real line as shown in the figure, how one can argue or prove that $b$ is on the right side of $a$ or $a$ is on the left side of $b$?
2
votes
4answers
56 views

Split the set of real numbers into $2$-element sets

How can we split $\mathbb{R}$ into disjoint sets, each consisting of $2$ elements? I have found a similar (though much more general) question here. But I am unable to deduce an answer to my specific ...
0
votes
1answer
39 views

Real number system

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot ...
5
votes
2answers
93 views

A question about infinities and pots of paint

This question is inspired by http://math.stackexchange.com/a/1052384/66307 and quotes from it heavily. Take a countably infinite paint box; this means that it has one color of paint for each positive ...
2
votes
1answer
34 views

Justification of steps

I'm taking Real Analysis starting in January and I'm getting a head start now so as to make the class somewhat easier. In the text we are using (Bartle and Sherbert, 4th ed.), it has examples where ...
32
votes
12answers
3k views

Why are integers subset of reals?

In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). ...
0
votes
2answers
93 views

If $a,b,c$ are positive real numbers, then $\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$ [closed]

If $a$, $b$, and $c$ are positive real numbers such that $abc=1$, then prove that $$\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ca+1}\geq \frac{3}{2}$$ Progress I think the relevant concept would ...
4
votes
1answer
85 views

How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$?

How to prove that $\sqrt{15}-\sqrt{8}+\sqrt[3]{7}<3$ without using calculus/high school methods or routine calculation of the roots with the precision needed? The difference between them is close ...
0
votes
1answer
23 views

Can we find a natural number $m≠t$ verifying $[A^{c^{t}}]=[A^{c^{m}}]$ [closed]

Let $t$ be a non natural namber. My question is: 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 $A>1$ ...
0
votes
1answer
42 views

Cantor Intersection Theorem Without Closedness, counterexample

The Cantor Intersection Theorem is that Let $\{S_1,S_2,S_3,...\}$ be a countable collection of nonempty sets in $\mathbb R$ such that: $S_{k+1} \subset S_k$ for $k=1,2,3...$ Each $S_k$ ...
2
votes
1answer
42 views

How I can solve this equation with respect to the variable $t$?

How I can solve this equation with respect to the variable $t$? $$\left\lfloor{\frac{\ln(t+1)}{\ln 2}}\right\rfloor=\left\lfloor{\frac{\ln t}{\ln2}}\right\rfloor+1$$ where $\left\lfloor {y} ...
2
votes
2answers
60 views

Prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent

How do I prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent for every x in real numbers except for $x=0$? I tried using the ratio test, but it ...
0
votes
3answers
64 views

Is$[0,1] \left\backslash\right. \left\{ 1/n :n \in \mathbb Z^+ \right\}$ compact if given the subspace topology?

Let $[0,1] \left\backslash\right. \left\{ 1/n :n \in \mathbb Z^+ \right\}$ be given the subspace topology. Is it compact or not ?
3
votes
0answers
43 views

Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
1
vote
1answer
53 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, ...
1
vote
3answers
45 views

Infimum of $\left\{\frac{n}{n^2+1}\:\:;\:n\:\in \mathbb N\right\}$ with a proof

Consider $A=\left\{\frac{n}{n^2+1}\:\:;\:n\:\in \:N\right\}$. I need to find and prove $\inf(A)$. So I know that I need to prove that for every $\epsilon > 0$ exists some $a\:\in A$ such that ...
2
votes
4answers
57 views

Proof that $ \forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0 $

How to proof simply that $$ \forall x,y \in \mathbb{R} \qquad x^2+y^2+(x-1)(y-1)>0 $$
1
vote
2answers
29 views

How to specify each digit of a real number in decimal representation in set theory?

So real numbers have decimal representations. If you want to say the $n$th digit of some real number, how do you say this formally in set theory?
6
votes
1answer
489 views

Does any one-to-one function exist that satisfies this inequality for all real numbers?

Does there exist a one-to-one function $f: \Bbb R \to \Bbb R $ such that $f(x^2) - (f(x))^2 \geq \frac 1 4\ \ \forall x \in \Bbb R$ ? I've tested this with many one-to-one functions but the ...
2
votes
1answer
41 views

Prove that the following set is dense in R

I need to show $ S = { m\cdot \sqrt{2}+ n\cdot \sqrt{3},where~m,~n~in~\mathbb{Z}} $ is dense in $\mathbb{R}$. I showed that S has an element in $(0,ε)$ for every $ε>0.$ How do I proceed to show ...
0
votes
2answers
28 views

Proof of inequality of sums

I have the following to prove, with induction and any help would be appreciated! :) $n\in \mathbb{N}, \quad \left(\, x^{1},\ldots,x^{n}\,\right)\in\mathbb{R}^{n}$ $$ \left(\,\sum^{n}_{i\ =\ ...
2
votes
3answers
71 views

Inequality proof for $1+x^3\geq x+x^2$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. I tried it based on previous one, but without success... The prev. ...
0
votes
2answers
20 views

Proof of an inequality in $\mathbb{R}$

I have an inequality to prove and I can't get a hold of it... I hope someone can help with it or point me in the right direction. $x,y\in\mathbb{R},\quad \epsilon\in\mathbb{R}:\epsilon\not=0$ $$ ...
1
vote
0answers
13 views

real number and continuity

I have just read Courant's Introduction to Calculus and Analysis. What makes me confusion is the section "Real Number and Nested Intervals". In the Postulate of Nested Intervals or the axiom of ...
0
votes
1answer
39 views

Order Axiom explanation

Can someone explain to me how to go about solving this axiom. I don't fully understand axioms, so I'm having problems. Let $x, y, z$ belong to the real number system and let > be an order operation ...
0
votes
2answers
57 views

$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon$>0. How to prove $x \le y$? [closed]

$x,y \in \mathbb R : x \lt y + \epsilon : \epsilon \gt 0$ Then prove $x \le y$ .
2
votes
3answers
154 views

$1^x = 1^y$ and $x,y$ belongs to Real Numbers.

$1^x = 1^y$, and $x,y \in \mathbb{R}$. Following the rule, same base has powers equal every $x$ should be equal to every $y$. $$1^x = 1^y$$ $$x = y$$ What went wrong?
0
votes
3answers
35 views

Is supremum part of the set or it is the bigest element out of it?

"If $\sup A$ is in $A$, then is $\sup A$ called also Maximum: $\max A$." So that means that $\sup A$ can be outside the set $A$? And lastly the upper barrier(or bound, not sure) is from $A$, ...
2
votes
1answer
28 views

Prove the set of transcendental number is dense in $\mathbb{R}$ using Baire category theorem

I want to show the set of transcendental number $T$ is dense in $\mathbb{R}$ using Baire category theorem. The fact is easier to show directly using definition and Archimedean property. But I was ...
3
votes
0answers
39 views

Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that accomplish this proposition

Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that $$|x- \frac{m}{n} |<\frac{1}{n^2}$$ It's clear that $-1<1/n^2 ...
3
votes
0answers
39 views

How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
7
votes
5answers
364 views

difference between nonpositive and negative numbers?

I am wondering if there is any difference between non-positive and negative numbers? I think that negative numbers mean "negative real numbers" and "Non-positive numbers" are negative real numbers ...
1
vote
4answers
93 views

Proof: For every $x \in \mathbb R\,\exists n\in\mathbb Z,\,r\in[0,1):x=n+r.$

I still do not understand how to approach proofs. Any help would be appreciated. For every real number $x$, there exists an integer $n$ and a real number $r\in [0,1)$ such that $x = n + r$. ...
3
votes
2answers
141 views

Percentage of rational numbers on an interval

Today, I just came up with this random question: What is the percentage of of rational numbers on an interval? Let $\mathbb{Q}$ be the set of the rational numbers: 1- Take an interval on the real ...
2
votes
1answer
37 views

Is there an orderpreserving bijection from $\mathbb R$ to $\mathbb R-\{0\}$?

Looking at $(\mathbb R,<)$ where $<$ denotes the usual order I understand that $\mathbb R-\{0\}$ will inherit an order (as every subset of $\mathbb R$). I could not find essential differences ...
1
vote
0answers
60 views

Normalizing Vectors to get short numbers

$\vec{A}$ is vector agent, $\vec{O}$ is vector Object, $m$ is a constant integer. The following expression is repeated with a different O for every loop cycle: ...