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

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0answers
23 views

Tell whether the given statement is true or false. Explain your choice. [on hold]

All whole numbers are rational numbers & No irrational numbers are whole numbers
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3answers
55 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
1
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1answer
43 views

Absolute Value Property of Field of Real Numbers

I don't think my thought process is correct. Also, does 'if and only if' indicate that I should automatically resort to proof by contradiction? Show that ${|b|} \le {a}$ if and only if $ {-a} \le b ...
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1answer
46 views

Check dedekind cut for root 2

root 2 corresponds to the Dedekind cut $(A,B)$ where: $A = \{ x \in \Bbb Q|x\lt 0$ or $x^2\lt 2 \}$ $B = \{ x \in \Bbb Q|x\ge 0$ or $x^2\ge2 \}$ Check that this is a Dedekind cut of $\Bbb Q$ ...
0
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1answer
35 views

Conventions adopted for extended reals

It is known that $0^0$ despite being an indeterminate limit form, is usually defined to be equal to $1$. I wonder whether similar conventions exist for some other "indeterminate forms" in the context ...
0
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0answers
28 views

Is there a name for complex numbers over affinely extended reals?

Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis ...
1
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1answer
55 views

What is the relation between $\operatorname{int}f(S) $ and $ f(\operatorname{int}S)$ & $f(\overline S)$ and $ \overline {f(S)} $

$\newcommand{\int}{\operatorname{int}}$Let $ f: \mathbb R→ \mathbb R $ be a continuous function and let $S$ be a non-empty proper subset of $\mathbb R$ . Which one of the following statements is ...
1
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1answer
40 views

How to calculate the cardinality of the complement of two countable sets of reals?

Let $A,B\subseteq\Bbb R$ be countable sets. Denote by $A'$ and $B'$ the complements (in $\Bbb R$) of $A$ and $B$ respectively. What is the cardinality of $C=A'\cap B'$? I cant figure this ...
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2answers
50 views

Existence of $n$th roots in $\mathbb{R}^+$

I am trying to understand this proof. Theorem. If $a$ is a nonnegative real number and $n$ is a positive integer, there exists a real number $b\geq 0$ such that $b^n=a$. Proof. Let ...
0
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1answer
16 views

Hilbert Symbol over $\mathbb{R}$ (bilinearity)

Let $\mathbb{R}$ be the field of the reals and let $a,b,c \in \mathbb{R}^{\times}$. As you probably know, the Hilbert symbol over any field $K$ is defined as: $$(\frac{a,b}{K}) = 1 \text{ if } \exists ...
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0answers
45 views

Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
3
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2answers
123 views

Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.

Basically I need help in proving that if $U\supseteq \mathbb Q $ is an open set in $\mathbb R$ with the usual topology then $\mathbb R \setminus U$ is countable. I'm not really sure how to proceed. ...
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2answers
72 views

Topological properties that the real line does not have

The following question is kind of strange, but I would like to know what topological properties $\mathbb{R}$ (with the standard metric topology) does not posses? I know this question sounds a bit ...
0
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2answers
32 views

Proving that the supremum of $(a,b)$ is $b$.

I was trying to prove it this way. Let $a,b\in\mathbb{R}$ with $a<b$. Let $ I=(a,b)$. As for all $x\in I$,$x<b$, we get $b$ is an upper bound. let $u=\sup I$, then $u\leq b$. If $u=b$ it's done. ...
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
3
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1answer
75 views

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$?

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
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1answer
24 views

Representation of real numbers under positional systems

Consider the representation of a real number in the base $\beta \geq 2$ as the string $$ (-1)^\sigma(b_nb_{n-1}\ldots b_0.b_{-1}b_{-2}\ldots)_{\beta} \, , $$ where $b_n, b_{n-1}, \ldots$ are ...
0
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1answer
17 views

standard notation to handle representation of a real number on a computer

Is there a standard notation to handle the effective representation of some real number $x$ on a finite machine ? I have in mind some kind of braces, but I am not sure it is appropriate. Let me try to ...
-2
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2answers
124 views

Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? [closed]

Here is a small intuition why it should be the later. Let $\omega$ be the number of all natural numbers. Then what is the smallest real number? We can write reals in binary form. Usual logic would ...
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2answers
34 views

Uniqueness of sum of exponentials

I would like to know if there is an example of two non-trivial sets of real numbers (for the definition of "non-trivial" see below) $X=\{ x_1, \ldots x_n \}$ and $Y = \{ y_1, \ldots y_m \}$, with $m$ ...
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2answers
37 views

Definition of a Dedekind cut help?

So the definition of a Dedekind cut is: a = sup{r $\in$ $\mathbb{Q}$: r < a} for each a $\in$ $\mathbb{R}$. Subsets a $\in$ $\mathbb{Q}$ having the form {r $\in$ $\mathbb{Q}$ : r < a} satisfy ...
2
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2answers
39 views

Is “the nth root of x” well-defined without further qualification?

(No, I'm not asking if $\sqrt{-1} = +i$ or if $\sqrt{-1} = -i$. Yes, I know $+i$ is the principal square root.) Consider the cube root of -8. If asked to evaluate it, I would say -2, and I think we ...
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1answer
30 views

Weierstrass M- test for real valued functions

Let $x \in [0,\infty)$ and $\sum_{n=1}^{\infty}\frac{x^{2}-nx}{n^{3}+nx}$. I want to see whether the series converges uniformly on $[0,\infty)$ . Using the M-test we have ...
1
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1answer
60 views

Extending the complex numbers by the solution of $|x| = -1$

I don't think I've ever encountered a situation where I've wanted to solve equations of the form $|x| = -1$, but you often hear that mathematics should be explored for the sake of mathematics. I'm ...
2
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4answers
83 views

If $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function?

Like the title reads, if $f(\alpha x, \alpha y) = f(x,y)$ is $f$ some special function? Assume $x,y,\alpha\in\mathbb{R}$.
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1answer
22 views

$\limsup_n a_n + \limsup_n b_n \le \limsup_n c_n + \limsup_n d_n$ if $a_n+b_n=c_n+d_n$ and $a_n$ maximum

I already posed some questions on inqualities between superior limits of real sequences, so here there is another one: Let $(a_n)_{n\ge 1}, (b_n)_{n\ge 1}, (c_n)_{n\ge 1}$, and $(d_n)_{n\ge 1}$ be ...
1
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1answer
25 views

Theory behind prime generating function $p=an+b$, where $a, b$ are real coefficients

The Green–Tao theorem states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words there exist arithmetic progressions of primes, with k terms, where k ...
3
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2answers
68 views

How to define the operation of division apart from the inverse of multiplication?

Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am ...
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2answers
46 views

Prove result of xy [closed]

If $$25^x = 7\quad \text{and}\quad 7^y = 125$$ then $xy=\frac{3}{2}$. Can someone explain me why $xy$ is equal to $\frac{3}{2}$? Thank you
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0answers
16 views

Please gauge my understanding of this-A proof from GH hardy's Book

What i did to understand this was i took took a unit length $A_{1}A_{2}$. And BC=0.4 such that it lies inside it. Taking k=3. The line segment $A_{1}A_{2}$ IS divided into 3 equal parts say $p_1$, ...
5
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2answers
40 views

How to prove the not-so-long rays are homeomorphic to the reals?

The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable ...
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0answers
18 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
3
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1answer
97 views

Can we prove that set of irrational numbers is a set using Zermelo-Fraenkel axioms?

To remove paradoxes of naive set theory, We started with the axioms of Zermelo-Fraenkel and developed a set theory. Where we are building sets starting from a empty set. How to construct set of ...
3
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2answers
33 views

Real Analysis. Bounds, Infimum and Supremum

Given that $S=\left\{x \mid 4x^2 > x^3 + x\right\}$. (1) Determine whether $S$ is bounded. (2) Determine their supremum and infimum. I divided the equation by $x$ to have a quadratic. Then my ...
2
votes
1answer
71 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 ...
3
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9answers
262 views

How is $x \leq x^2$ false? [closed]

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
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1answer
64 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 ...
5
votes
4answers
146 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 $x, y \in \mathbb{R}.$
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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 ...
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1answer
53 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
19 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 ...
5
votes
1answer
90 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
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2answers
25 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
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3answers
54 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 ...
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1answer
54 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
64 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
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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 ...
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2answers
79 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
364 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 ...