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

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2
votes
4answers
55 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
27 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
476 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
38 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
27 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
70 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
36 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
135 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
33 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
26 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
37 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
35 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
351 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
89 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
138 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
36 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
58 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: ...
1
vote
0answers
17 views

Are numbers that are already palindromes lychrel numbers?

I'm working on a Project Euler problem where I'm supposed to determine lychrel numbers. I'm not sure I understand what a lychrel number is. From my understanding a lychrel number is any number that ...
0
votes
0answers
20 views

How to Prove a Metric Space is Sequential? [duplicate]

Let's say I have a space X with a "d-metric" on X, a function d:X×X→R that has the following 2 properties: d(x,y)≥0 for all x, y∈X and d(x,x)=0 for all x, y∈X. The d-ball of radius r centered at x ...
0
votes
1answer
29 views

Showing R is Sequential

I want to prove rigorously that R is sequential. Sequential means that every sequentially open set is open (see: http://en.wikipedia.org/wiki/Sequential_space). I can understand intuitively why R is ...
1
vote
1answer
65 views

Prove it's Sequential and not First Countable

Let's say I have a a space X, the quotient space of R (the reals) obtained by identifying all points of Z (the integers). How do I prove that X is sequential but not first countable? (sequential ...
1
vote
1answer
56 views

A field between $\mathbb{Q}$ and $\mathbb{R}$ ?

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
0
votes
1answer
43 views

How do I show R is regular?

How do I prove that R (the reals) are a regular space? This means I need to show that given any nonempty closed set F and any point x that does not belong to F, there exists a neighborhood U of x and ...
-1
votes
1answer
23 views

How I can simplify this expresson? [closed]

How I can simplify this expresson? $$v=((2^{p+q}-1)/(2^{p}-1))$$ where $p,q$ are natural numbers.
0
votes
0answers
87 views

If $ax + by ≤ bx + cy ≤ cx + ay$ then $b ≤ c$

Let $a, b, c, x, y$ be positive real numbers such that $$ax + by ≤ bx + cy ≤ cx + ay$$ Prove that $b ≤ c$. As of now I did simple algebraic manipulation: I just put all the $a$'s on one side ...
0
votes
1answer
27 views

Systems of equations and product of polynomials in linear algebra

I'm scratching my head trying to represent and solve this 2 problems using linear algebra. Almost immediately, any book I'm aware of, introduces vectors as a building block for matrices which in turn ...
-3
votes
1answer
37 views

Proof by mathematical induction [closed]

Let $x$ be an arbitrary real number. By mathematical induction it is requested to prove that for arbitrary natural number $n$ applies: $$\sqrt{x^2+\sqrt{x^2+\dots+\sqrt{x^2}}}<|x|+1$$ The square ...
4
votes
3answers
125 views

Is there a real number $r$ such that $\sum\limits_{k=0}^{\infty}\frac{p_k}{r^k}=e$?

Let $p_n$ denote the sequence of prime numbers, with $p_0=2$. I'm looking for a real number $r$ such that $\sum\limits_{k=0}^{\infty}\frac{p_k}{r^k}=e$. It's easy to show that $r>5$, with ...
-4
votes
3answers
49 views

How to prove that the intersection of $\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$ is $\mathbb Q$? [closed]

Let $S=\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $T=\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$. Prove that $S \cap T = \mathbb{Q}$. We are currently studying proofs by contradiction, but I don't ...
0
votes
1answer
33 views

Let $q \in \mathbb{Q}$ and $x \in \mathbb{R}-\mathbb{Q}$. Prove that $q+x \in \mathbb{R}-\mathbb{Q}$

This is a homework problem for my Real Analysis course and I am having trouble getting started in the right direction. I understand the definition of the set of rational numbers and how ...
6
votes
2answers
75 views

What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)?

What is the topological properties of $\mathbb R $ that makes it uncountable (as compared to $\mathbb Q $)? Further, what axioms (or properties) of $\mathbb R $ do these topological properties depend ...
0
votes
2answers
36 views

How I can estimate or calculate the integer part of $b/a$ [closed]

Let $a$ be a natural number and $b$ a real number (not a natural number) such that $b>a$. How I can estimate or calculate the integer part of $b/a$
0
votes
1answer
13 views

About the the fractional part

I want to prove this result: If $a+b$ is an integer and $0<b<1$ implies that the fractional part of $a$ is just $1-b$.
7
votes
1answer
74 views

Irrationals in Cantor Set

It is well known that the Cantor set is uncountable. Hence it contains irrationals. What are the 'nice' irrationals in the Cantor set. Here, I am expecting irrational numbers in the form of square ...
0
votes
0answers
34 views

What's the solution set $S \subset \mathbb{R}^2$ of this equation?

I see that $(1,1)$, $(2,4)$ and $(4,2)$ are in $$S= \{(x,y) \in \mathbb{R}^2: \, x^y = y^x\}$$ My question is: The set $S$ contains many others elements? Thanks for any suggestions and helpful ...
1
vote
1answer
46 views

How I can calculate the fractional part of this number

Let us consider this real number: $$z=\frac{(-\sqrt3+2)^{2^{p-2}}+(\sqrt3+2)^{2^{p-2}}}{2^{p}-1}$$ My question is: How I can calculate the fractional part of this number for big natural number $p$. ...
3
votes
4answers
605 views

About the location of natural numbers

My question is concerned on the location of natural numbers: Find sufficient and necessary conditions on two real numbers $a$ and $b$ such that the open interval $(a,b)$ contain at least one natural ...
2
votes
2answers
41 views

Prove that $\frac{301}{900} $ number is $\sup$ of this set

Problem Prove that $\frac{301}{900} $ is $\sup$ of this set: I have the following set, $$S=\{ 0.3 , 0.33 , 0.334 , 0.3344, 0.33444,... \}$$ I don't know how to do it, I know that I can ...
0
votes
2answers
20 views

Find two real numbers $a$ and $b$ such that $a<q<b$ and $b-a<1$

Let $q$ be a natural number. Find two real numbers $a$ and $b$ such that $$a<q<b$$ and $$b-a<1$$ Can we deduce something about the nature of $a$ and $b$ such as rationals or irrationals.
2
votes
1answer
30 views

Baby Rudin Problem Chapter 2, Problems 17(c) and (d)

Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Then I've managed to show that (a) $E$ is not countable, and (b) $E$ is not dense in $[0,1]$. ...
0
votes
1answer
20 views

Interesting real-valued function problem

Find all real-valued functions $f(m),m\in \mathbb R$ such that for $\forall x,\forall y \in \mathbb R$ $$f((x-y)^2)= (f(x))^2 - 2xf(y) + y^2.$$ I tried several ideas, one of them is taking special ...
0
votes
1answer
32 views

Interior points/Boundary points of sets

I need a little help understanding exactly what an interior & boundary point are/how to determine the interior points of a set. If you could help me understand why these are the correct answers or ...
2
votes
3answers
137 views

How are the elementary arithmetics defined?

In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that ...
0
votes
2answers
46 views

An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$

Problem: Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon ...
0
votes
1answer
32 views

Rudin Question Help

Given a set $A$ that has an upper bound,may we define $\sup A$ as $$\sup A = \max_{w \in A} \{\lim_{ \epsilon \to 0} w + \epsilon \}$$ Is this equivalent? Problem (7) Fix $b>1$, $y>0$ and ...
-6
votes
2answers
46 views

What is the number of natural numbers between $a$ and $b$ [closed]

Let $a<b$ two real numbers. So, what is the number of natural numbers between $a$ and $b$.
1
vote
2answers
16 views

Show that there exists two real numbers $a,b,q$ such $q<a≤r≤b<q+1$

Let $r$ be a rational number and not an integer. Show that there exists two real numbers $a$ and $b$ and an intger $q$ such $$q<a≤r≤b<q+1$$ Of course I would like that $a,b,q$ depends on $r$.