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

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2
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0answers
32 views

Dedekind Construction Of Real Numbers

If we define Dedekind-real numbers as Dedekind cuts, i.e. $\sqrt 2 = \{\text{rationals less than }\sqrt2\} \cup \{\text{rationals more than } \sqrt2\}$, can we define addition and multiplication of ...
2
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9answers
201 views

How is $x \leq x^2$ false?

There's an equation that says $$x \leq x^2$$ and $x \in \mathbb R$. What I can solve and clearly see is that this equation would be true for any value of '$x$' but then how come my maths teacher ...
5
votes
1answer
52 views

Approximating nice functions with wild ones

Let $X$ and $Y$ be toplogical spaces, and call a function $f:X\to Y$ wild if the preimage $f^{-1}(\{y\})$ is dense in $X$ for every $y\in Y$ -- or, equivalently, if the image of every nonempty open ...
4
votes
4answers
125 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
3
votes
3answers
102 views

Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true?

I'm having some trouble deciding whether this inequality is true or not... $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ for $x, y \in \mathbb{R}.$
2
votes
1answer
44 views

What would be interesting maps to use on that Eudoxus reals?

I'm trying to understand Eudoxus Reals. From wikipedia: Let an almost homomorphism be a map $f:\mathbb{Z}\to\mathbb{Z}$ such that the set $\{f(n+m)-f(m)-f(n): n,m\in\mathbb{Z}\}$ is finite. We say ...
-4
votes
0answers
55 views

I have problem on this. [closed]

Find a number $M$ such that $$|x^3 - x^2 + 8x|\le M$$ for all $$- 2\le x\le 10.$$
-6
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1answer
51 views

Cauchy Schwartz Inequalities [closed]

$| x^{2} - 9x + 1 |\leq M$ for all $- 1\leq x \leq 5$ find M
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0answers
17 views

Dimensional reduction of system of ODEs

Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers: $$dX_1/dt = \ldots\\ dX_2/dt = \ldots \\ \ldots ...
1
vote
0answers
60 views

Best intro to Fundamentals of Mathematics? [closed]

If you like mathematics it's likely that you also want to have the most solid foundations in number theory and analysis possible. I have just finished Elliott Mendelson's book Number Systems and the ...
5
votes
1answer
88 views

Solve $a^x+b^x=c$ for $x$

I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$). I know this admits a solution (details below), but it's such ...
1
vote
2answers
22 views

What is the difference between maximal element and least upper bound?

Maximal element is given as: Let $(P,\leq)$ be a partially ordered set and $S\subset P$. Then $m\in S$ is a maximal element of $S$ if for all $s\in S$, $m \leq s$ implies $m = s$. Least upper ...
4
votes
3answers
49 views

Topology: Prove that this subspace topology is discrete

The question is from Topology and Its Applications Chapter 1, by William F. Basner. The question states the following, Let $\mathbb{Z}$ be a topological space with subspace topology inherited from ...
0
votes
1answer
48 views

Proof that the Order topology on $\mathbb{R}$ has the same basis as the Euclidean topology

I want to prove that the Order topology on $\mathbb{R}$ has the same basis as as the Euclidean topology on $\mathbb{R}$. Assume that the only thing we know about the order topology is that it has the ...
2
votes
2answers
62 views

Non-linear system of equations

Solve following system of equations over real numbers: $$ x-y+z-u=2\\ x^2-y^2+z^2-u^2=6\\ x^3-y^3+z^3-u^3=20\\ x^4-y^4+z^4-u^4=66 $$ This does not seem as hard problem. I have tried what is obvious ...
0
votes
2answers
21 views

Problem with one of the order property proofs

I came across an exercise in an analysis book that requires me to show that if $A$ is a real number such that $0 \leq A \leq B$ for every $B>0$, then $A=0$ What I fail to understand here is if ...
1
vote
2answers
70 views

Nature of the Continuum

When we construct the vector space $\mathbb{R}^n$ we state that every element $x$ is a limit of some rational Cauchy sequence $\{x_i\}$. Two Cauchy sequences $\{a_i\}$ and $\{b_i\}$ are equal if they ...
3
votes
3answers
339 views

Symbol for set of strictly positive real numbers?

Is there any standard symbol for the set $\{x\in\mathbb{R} : x > 0\}$? I think $\mathbb{R}^{+}$ usually includes zero. Some sources say I should use $\mathbb{R}^{*}_{+}$ but it looks slightly ...
1
vote
1answer
86 views

Example of a homeomorphism on the real line?

I'm to give a short presentation on "basic topology" for a first semester undergrad analysis course. Naturally the professor does not expect me to master the topic, so I'm just trying to get some of ...
5
votes
1answer
88 views

Could Euclid have proven that multiplication of real numbers distributes over addition?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
7
votes
1answer
117 views

Could Euclid have proven that real number multiplication is commutative?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
2
votes
0answers
65 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
10
votes
0answers
108 views

If $n^x\in\Bbb Z,$ for every $n\in\Bbb Z^+,$ then $x\in\Bbb Z$ [duplicate]

Let $x$ is a real number such that $n^x\in\Bbb Z,$ for every positive integer $n.$ Prove that $x$ is an integer. I got that problem here and it looks difficult, I tried writing $x$ as $\lfloor ...
1
vote
1answer
96 views

(Ir)rationality of Real Numbers

I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right? For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational ...
1
vote
0answers
19 views

Expressing $y=\lfloor rx\rfloor$ in PA

The formula: $$y=\lfloor x\sqrt2\rfloor$$ is expressible in first-order PA, as: $$y^2<2x^2<(y+1)^2$$ So, even though $\sqrt2$ isn't a natural number, we can still represent a formula with ...
3
votes
1answer
35 views

Dedekind cuts and multiplication

A common way to define multiplication for Dedekind cuts is to first define it for pairs of positive reals, and to then extend it to general pairs of reals case by case. Is there an alternative ...
2
votes
2answers
59 views

$x_1,x_2,…,x_n$ are positive real numbers, $\sum_{i=1}^n x_i^2 = 1$, to find the minimum value of:

$$\sum_{i=1}^n \frac{x_i^5}{s - x_i}$$ with $$s = \sum_{i=1}^{n}x_i$$ I used Cauchy Schwarz inequality: $$(\sum_{i=1}^n \frac{x_i^5}{s - x_i})(\sum_{i=1}^{n}\frac{s-x_i}{x_i}) \geq 1$$ ...
4
votes
3answers
47 views

$a_1,a_2,…,a_n$ are positive real numbers, their product is equal to $1$, show: $\sum_{i=1}^n a_i^{\frac 1 i} \geq \frac{n+1}2$

it says to use the weighted AM-GM to solve it, because the inequality is not homogenous I've tried to use $$\lambda _ i = \frac{a_i^{\frac1i -1}}{\sum_{k=1}^n a_k^{\frac1k -1}}$$ this $\lambda$ is ...
2
votes
1answer
30 views

$a_1,a_2,a_3,b_1,b_2,b_3$ are positive real numbers, show: $\sqrt[3]{(a_1+b_1)(a_2+b_2)(a_3+b_3)} \geq \sqrt[3]{a_1a_2a_3} + \sqrt[3]{b_1b_2b_3}$

The question says one only needs the AM-GM inequality, I've been stuck here for more than one hour. $$(a_i + b_i) \gt a_i$$ and $$a_i + b_i \gt b_i$$ therefore, $$ ...
1
vote
1answer
26 views

On the addition of elements in Cantor Set

Let $C$ denote the standard Cantor set. It is well known that $C+C=[0,2]$. (Here, by $C+C$ we simply mean what it should be naturally: the set $\{x+y\colon x,y\in C\}$. Question 1. For some $a$ in ...
4
votes
3answers
734 views

Does same cardinality imply a bijection?

This came up today when people showed that there is no linear transformation $\mathbb{R}^4\to \mathbb{R}^3$. However, we know that these sets have the same cardinality. I was under the impression ...
2
votes
3answers
95 views

Is there any difference between 'all real numbers' and '$(-\infty, \infty)$'

I've just thought about this. All the textbooks I've been looking at for pre-calc, the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as ...
2
votes
1answer
29 views

Is the set of real numbers in $[0,1]$ with digits $1$ and $3$ only in their development in base $5$, dense in $[0,1]$?

Let $E$ denote the set of real numbers in $[0,1]$ with digits $1$ and $3$ only in their development in base $5$. How to prove that $E$ is dense in $[0,1]$? Is this the right way to see that E is ...
1
vote
1answer
57 views

A problem in Algebra [duplicate]

If $a+b+c+d=0=a^7+b^7+c^7+d^7$, Then prove $a(a+b)(a+c)(a+d)=0$ and all are real numbers. I am confused and don't even know which to tag for help.
4
votes
0answers
40 views

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
10
votes
5answers
387 views

Foundation of ordering of real numbers

This might be a silly question, but what is the mathematical foundation for the ordering of the real numbers? How do we know that $1<2$ or $300<1000$... Are the real numbers simply defined as ...
1
vote
1answer
30 views

Is a real closed, bounded interval a locally compact Hausdorff space?

Does this hold? I've been confused by the statement of the Riesz-Markov-Kakutani representation theorem; that is, the formulation is as follows: Let $X$ be a locally compact Hausdorff space. For ...
2
votes
2answers
44 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
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votes
0answers
29 views

Definition of isometries [duplicate]

We have been learning about isometries and how reflections, translations etc. and how they can affect a function. I was wondering if someone could help me with this proof by using the definition of ...
3
votes
5answers
138 views

Why can a closed, bounded interval be uncountable?

From what I have read, all finite sets are countable but not all countable sets are finite. As I understand it, Countably Finite --- a one to one map onto $\Bbb{N}$ with a limited number of members ...
0
votes
0answers
18 views

Limit of ratio of consecutive terms equals lmit of $n^{th}$ roots? [duplicate]

I got the following solved exercise from an old tutorial sheet. Evaluate the limit $$\lim_{n \rightarrow \infty} \Bigg{(}\frac{(2n)!}{(n!)^2}\Bigg{)}^{\frac{1}{n}}$$ So here is now the solution ...
0
votes
0answers
73 views

Set Theory (Real Numbers)

I have seen in a book that a number whose square is nonnegative is called real number. How can we explain what a real number is?
4
votes
1answer
112 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
0
votes
3answers
40 views

Possible values of 'a' ? $f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$

If $$f(x)=(x^2+ 2 ax +a^2-1)^{\frac{1}{4}}$$ has its domain and range such that their union is set of real numbers,then what should be the possible values of a? What can be the approach?
5
votes
3answers
138 views

Question about required rigour with intro real analysis text

I have just begun trying to self study introductory analysis and I am just having some questions about being specific on rigour. In the book I am using, titled Introduction to Real Analysis, 4th ...
3
votes
5answers
266 views

Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number?

Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ? Thank you for any help
1
vote
1answer
55 views

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...
8
votes
5answers
1k views

Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
3
votes
2answers
41 views

Cantor Sets in perfect sets in the Real numbers

My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets. In one blog a read a proposition. ''Every perfect set ...
4
votes
1answer
43 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...