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2
votes
1answer
13 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
0
votes
1answer
39 views

Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$… show $f$ preserves order.

Spivak Calculus, 4th ed., problem 3-17: If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
1
vote
2answers
42 views

Dedekind cuts and circularity

Can someone help me understand the concept of Dedekind cut? I'm having trouble understanding it because it appears that in order to understand what Dedekind cuts are I already must have a very ...
2
votes
1answer
24 views

Hausdorff dimension of $\mathrm{R}^d$

I assume the Hausdorff Dimension of $\mathrm{R}^d$ is $d$.. To prove this I guess one hast to prove these two statements: the $\alpha$-dimensional Hausdorff measure of $\mathrm{R}^d$ is $0$ for $d ...
0
votes
1answer
19 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
0
votes
2answers
28 views

What does P(A U B) mean, in terms of real values?

I can't find a proper summary or reference of how to translate formulas in probability notation to arithmetic notation (i.e. when using real values). For example, if $P(A) = .7$ and $P(B)=.35$, what ...
3
votes
3answers
48 views

Besides being symmetric, when will a matrix have ONLY real eigenvalues?

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation ...
1
vote
0answers
13 views

Random Algebra Problem 2

Prove that if a, b, c are integers and x, y, z are non-integer real numbers and $\alpha$ is a real number, for every given set of x, y, z, the number $\alpha$ obtained from the following equation: ...
1
vote
1answer
16 views

Cardinality of rational exponentiation orbit space

Let $X=(0,\infty)$ be the set of positive real numbers. Let $G=\mathbb{Q}\backslash\{0\}$ be the multiplicative group of rational numbers. $G$ acts freely on $X$ by exponentiation: $r\cdot x=x^r$ for ...
2
votes
1answer
40 views

A sort of partition of real numbers

I'm looking for two injective functions $f, g:\mathbb R\to\mathbb R$ with $f(x)+g(x)=x$ for all $x\in\mathbb R$ and $\operatorname{Im} f\cap\operatorname{Im} g=\emptyset$. I've tried nothing and I'm ...
0
votes
2answers
13 views

Division question

Let Z be the number of 8-digit numbers with 8 different digits, none of which is 0. How many 8 digit numbers exist that are divisible by 9, that have 8 different digits, none of which is 0. Answer in ...
1
vote
0answers
39 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
3
votes
0answers
57 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
0
votes
1answer
26 views

Rewriting integrals over a symmetrical set

I have $\int_\mathbb{R}f(x)\cdot \mathbf{1}_{A}(x)\,dx$, with $f(x)$ integrable and $A=[-a,-b]\cup[b,a]$. I rewrote the integral as: $\int_{-a}^{-b}f(x)\,dx+\int_{b}^{a}f(x)\,dx$ but that is ...
1
vote
2answers
74 views

Are There Numbers Beyond the Reals?

I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers. So, if ordinal numbers are viewed as an extension of the ...
0
votes
0answers
25 views

About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in ...
1
vote
2answers
38 views

If an infinite subset of a set has a limit point in it, the set is compact

I'm working through Rudin right now, and I am stuck on one of his proofs. In theorem 2.41 he tries to prove that, if every infinite subset of a set $E$ contains a limit point in $E$, then $E$ is ...
1
vote
1answer
30 views

“$\leq_\Bbb{R}$” is restriction of “$\leq_{\overline{\Bbb{R}}}$” to $\Bbb{R}$ ?!

In definition of Affinely Extended Real Numbers, I think that "$\leq_\Bbb{R}$" is restriction of "$\leq_{\overline{\Bbb{R}}}$" to $\Bbb{R}$ Is it correct?... Thanks in advance!
0
votes
1answer
50 views

$\mathbb{R}$ written as union of open intervals

Is $\mathbb{R} = \bigcup \limits_{n=1}^{\infty} (-n,n)?$
0
votes
1answer
24 views

Proof: $a,b \in \overline{\Bbb{R}}$, $a\leq b < +\infty \wedge a \in \Bbb{R} \Rightarrow b \in \Bbb{R}$

I must to proof the following: Prop.: let be $a,b \in \overline{\Bbb{R}}$ then:$$a\leq b < +\infty \wedge a \in \Bbb{R} \Rightarrow b \in \Bbb{R}$$ Proof: by contradiction I have $b \notin ...
5
votes
2answers
44 views

Proving well definedness of addition in real numbers constrructed from cauchy sequences.

While studying real analysis, I got confused on the following issue. Suppose we construct real numbers as equivalence classes of cauchy sequences. Let $x = (a_n)$ and $y= (b_n)$ be two cauchy ...
2
votes
2answers
53 views

Does nth-roots with non-natural indexes exists?

My high school math teacher stated that roots with non-natural indexes are meaningless, just like $\frac{\infty}{\infty}$ or $0^0$, because "mathematicians decided so and so it is unless we change ...
2
votes
1answer
49 views

How do we decide which axioms are necessary?

I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings. Given some construction of the reals (e.g. Dedekind cuts or ...
3
votes
2answers
70 views

Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
1
vote
1answer
45 views

A question on completeness of $\mathbb R$ and countability of infinite sets

If $x \in \mathbb R$\ $\mathbb Q$ and $A$ is a set of real numbers such that $x \notin A$ and $\sup A=x$ , then do we necessarily have that $A$ is uncountable ?
0
votes
1answer
37 views

Simple algorithm Hermite Normal Form for 3x3

In the scope of the implementation of a model, I need to reduce a 3x3 real matrix into its Hermite Normal Form. I am very new to this kind of reduction and only find algorithm using complex notions ...
5
votes
2answers
69 views

Continuity of Real Number line

What is the property of real numbers that allows them to be seen as a continuous line, and how natural numbers, rational numbers lack this property?
2
votes
2answers
44 views

Determine real number exists for relation with square roots

We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$. I am to find whether a real number exists for this relation, and the real number that satisfies. I start by squaring both sides, which yields: $$x - 2 = 4x ...
2
votes
3answers
75 views

Who was the first person to construct the real numbers by Cauchy sequences in $\mathbb{Q}$

Who was the first person to construct the real numbers by Cauchy sequences in $\mathbb{Q}$? Was it Cauchy himself?
0
votes
1answer
48 views

Dedekind Cut additive inverse

Let $\alpha$ be Dedekind cut and define $\alpha^* :=\{x\in\mathbb{Q}|\exists r>0\space \text{such that} -x-r\notin\alpha\}$. I need to show that $\alpha^*$ is a Dedkind Cut and the additive inverse ...
4
votes
2answers
37 views

Find all the polynomials $p \in \mathbb R [X]$ such that $(x+1)p=(p')^2$

(Where $p'(x)$ is the derivative of $p(x)$) Research effort: what I thought is that given that $(x+1)|(p')^2$ then $(x+1)|(p')$ (I'd like to justify better this, but I don't know how) Then, ...
6
votes
0answers
105 views

Group Structure on $\Bbb R$

$(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ? Refer to ...
1
vote
1answer
63 views

Given a basis for $\mathbb{R}$, show that it constructs the standard topology on $\mathbb{R}$

Let $q_1, q_2, ...,$ be the rational numbers enumerated. Consider the countable collection $$\mathcal{B} = \{ B_{\frac{1}{n}}(q_i) \ | \ i,n \in \mathbb{N} \}$$ of open balls centered at rational ...
4
votes
1answer
59 views

About “Trichotomy Law for Real Numbers”

I must proof the "Trichotomy Law for Real Numbers": Prop. 1: let be $\Bbb{R}$ a complete ordered field, then $$\forall x,y \in \Bbb{R}(x=y \vee x < y \vee x > y)$$ Proof 1: by definition of $ ...
0
votes
1answer
55 views

f a real, continuous function, is it measurable?

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function. I need to show that is a measurable function. I tried working with the definition: Let $f: X \to \mathbb{R}$ be a function. If ...
2
votes
5answers
49 views

Given any real numbers $x \gt 0$ and $r \gt 1$ there exists $m \in \Bbb N$ such that $\frac 1 {r^m} \lt x$

I know it can be proven that given any real number $x \gt 0$ there exists $m \in \Bbb N$ such that $\frac 1 {2^m} \lt x$. I tried to generalise it but am surprisingly not getting anywhere. I ...
3
votes
1answer
55 views

Enough Dedekind cuts to define all irrationals?

Assuming that there are uncountably infinitely many irrationals between any two consecutive rationals, how can the Dedekind cuts (defined on the countably infinite rationals) define all the ...
1
vote
1answer
43 views

The distance between the integral multiples of two numbers

Given a positive real number $r \in \left(0,\infty\right)$, define $\left[r\right]$ to be the set of positive integral multiples of $r$: $$ \left[r\right] := \left\{nr\ :\mid\ n \in ...
0
votes
2answers
31 views

sum of dedekind cuts

The sum of Dedekind cuts alpha and beta is defined to be {a+b : a in alpha and b in beta}. But why does this correspond to the intuitive notion of sum? Why is sum defined this way? Spivak's book says ...
0
votes
1answer
16 views

One to one function behaviour

Like in pigeon hole principle , if one set of objects(S1) has more items than others set of objects(S2) and we try to fit that S1 in S2 ( that is mapping the values of S1 to S2 , we end up getting ...
0
votes
0answers
67 views

Prove that $\sup f(x) > \inf g(x)$

Let $f:D\rightarrow \mathbb{R}$ and $g:D\rightarrow \mathbb {R}$ be functions ($D$ nonempty). Suppose that $f(x) \le g(y)$ for all $x \in D$ and $y \in D$. Show that $\sup f(x) > \inf g(y)$ I ...
2
votes
2answers
92 views

Prove that $\sup f(x) \leq \inf g(y)$

Let $f: D \longrightarrow \mathbb{R}$ and $g: D\longrightarrow \mathbb{R}$ be functions ($D$ nonempty). Suppose that $f(x) \leq g(y)$ for all $x\in D$ and $y \in D$. Show that $$\sup f(x) \leq \inf ...
3
votes
2answers
67 views

$\alpha \in \mathbb{R}$ and $2^\alpha$, $3^\alpha \in \mathbb{N}$, implies $\alpha \in \mathbb{N}$?

Let $\alpha \in \mathbb{R}$. Suppose $2^\alpha$, $3^\alpha \in \mathbb{N}$. Does it implies that $\alpha \in \mathbb{N}$?
6
votes
11answers
2k views

Smallest next real number after an integer

This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not? Firstly, I know (I think) that $$\lim_{x\to \infty} ...
1
vote
2answers
57 views

Are these two expressions equal to each other?

Im working on a proof, and currently I'm trying to check some expressions to see if they are equal to each other. Using specific values as a test case, I get this expression But I can't tell if ...
0
votes
4answers
62 views

What does it mean for a function f(x) to be differentiable at x_0?

What does it mean for a function $f(x)$ to be differentiable at $x_0$? I need this to understand more concepts in real-analysis and calculus. Thank you.
1
vote
3answers
32 views

Real number multiplicative inverses expressed in another form

I've been asked to express the multiplicative inverse of $3 + \sqrt{5}$ in the form $c + d\sqrt{5}$, where $c,d$ are rational numbers. I understand that for some rational numbers $c,d$ we must have: ...
1
vote
1answer
63 views

What is cardinal of set of all Cauchy sequences?

Here are basically two questions. The first, what is the cardinal of equivalent Cauchy sequences of rationals? I know it's $\beth_1$ because of the set is essentially real numbers. But I want to know ...
0
votes
1answer
73 views

Axiom of choice and an example of a Well-ordered $\Bbb R$

From the axiom of choice we get that every set can be ordered in a way that will make it a well ordered set, including $\Bbb R$. However, since the ordinal of such a well-ordered set of $\Bbb R$ will ...
0
votes
1answer
40 views

Prove this statement? [duplicate]

I am having trouble with the following proof: Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \ge 9$. I have tried to directly ...