Tagged Questions

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

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1
vote
1answer
15 views

Find relations on the real number: transitive and/or antisymmetric

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$ antisymmetric and transitive antisymmetric and NOT transitive NOT antisymmetric ,but ...
1
vote
1answer
20 views

Find sufficient and necessary conditions in which $f(x)$ is a natural number

Let us consider the function $$f(x)=(-√3+2)^{2^{x-2}}+(√3+2)^{2^{x-2}}$$ where $x≥3$. We know for example that if $x$ is an integer, then $f(x)$ is also an integer. My question is: Find sufficient ...
1
vote
4answers
74 views

How to factor $x^{4}-22x^{2}+9$ over real numbers?

How do you factor $f(x) = x^{4}-22x^{2}+9$ over real numbers? I know that over integers it is $(x^2-4x-3)^2$.
6
votes
1answer
39 views

Can the order relation on $\mathbb{R}$ be recovered from the topology?

It's well-known that the usual topology on $\mathbb{R}$ is induced by the order relation, since the open intervals $\{(a, b) : a < b\}$ are a base for the topology. I am wondering if you can go ...
1
vote
1answer
58 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
0
votes
1answer
29 views

Show function achieves an absolute minimum and an absolute maximum [on hold]

Suppose f:R-->R is continuous and periodic with period P>0. That is, f(x+P)=f(x) for all x ∈ R. Show that f achieves an absolute minimum and an absolute maximum.
0
votes
1answer
25 views

Find an integer $n$ such that: $(2^{p}-2)!+1=n((-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}})$

Let $p$ be a prime number. My question is: Find an integer $n$ such that: $$(2^{p}-2)!+1=n((-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}})$$ I can write $$(-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}}=( ...
0
votes
1answer
19 views

How to prove that this enough to determine the structure of $\mathbb R$?

Let $F$ be a field and let $P\subset F$ such that the sets $P$, $\left\{ 0\right\} $ and $-P:=\left\{ -p:p\in P\right\} $ form a partition of $F$ and $x,y\in P\Rightarrow x+y\in P\wedge xy\in P$. I ...
1
vote
1answer
28 views

Let $p$ be a prime number. How I can simplify this expression

My question is: Let $p$ be a prime number. How I can simplify this expression: $$z=2∑_{j=0}^{2^{p-3}}C_{2^{p-2}}^{2j}2^{2^{p-2}-2j} 3^{j}$$ where $C_{2^{p-2}}^{2j}=((2^{p-2}!)/((2j)!(2^{p-2}-2j)!))$
0
votes
0answers
27 views

Unconstrained Optimal Control - $J = \frac{1}{2}x^2(2) + \frac{1}{2} \int_{0}^{2}(u^2 - 2xu)dt$

I've been given the following unconstrained optimal control problem, but I feel like I've made a mistake at some point. The system $\dot x = -x + u$, where u = u(t) is not subject to any ...
2
votes
3answers
70 views

Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$

The aim of this question is to show this lemma: Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$.
0
votes
1answer
19 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where ...
1
vote
0answers
39 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
0
votes
1answer
36 views

There exists a positive real number $u$ such that $u^3 = 3$

Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$. Show that there exists a positive real number $u$ such that $u^3 = 3$. So far, I have come up ...
0
votes
0answers
27 views

Prove using anxioms that $1 \neq 0$ - checking

Let $1=0$ we know that $x \cdot y=0$ if $x=0$ or $y=0$ so $1(1-1)=0 $ if $1=0$ or $1=1$ but $1=1$ is contradiction since we assume that $1=0$ so $1 \neq 0$ Is my prof correct ? If not how should it ...
0
votes
4answers
52 views

Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
1
vote
3answers
40 views

What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R ...
1
vote
1answer
24 views

Constant raised to the power of an even or odd function

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function. Would $a^{f(x)}$ be an even or odd function? And would $a^{g(x)}$ be an even or odd ...
1
vote
0answers
21 views

Constraining mathematics to a subset of $\mathbb{R}$

Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$. ...
3
votes
1answer
40 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
0
votes
2answers
50 views
1
vote
3answers
32 views

Use a function to represent positive real numbers?

Is it correct to define the positive real numbers as $\{f(x) = x^2\mid x \in \mathbb R\}$?
0
votes
0answers
23 views

How to find the general term of this sequence?

I would like to know if there is a way to find an explicit closed-form expression for the general term of the sequence (in $\mathbb{R}$) defined by $$ \begin{cases} a_0=1\\ a_{n+1}=1+\frac{1}{a_n}\ ...
1
vote
3answers
37 views

Calculate value of a real number, considering “n” as a natural number

How could I calculate the value of the real number: $$ (1 +i \sqrt{3})^n + (1 - i \sqrt{3})^n $$ ...considering $n$ as a natural number and $i$ as the imaginary unit.
0
votes
2answers
58 views

Show that the following statement is a theorem.

Suppose h is not a counting number and h is greater than 1, then there is a counting number n such that h is between n and n + 1. I am working through "Creative Mathematics" by H.S. Wall. The book ...
1
vote
2answers
38 views

Real numbers: powers inequality

Trying to prove the following inequality from textbook. Let $x>1$ be a real number, and let $q>0$ be a rational number. Then $x^q>1$. Let $p<q$ be a rationals numbers, and let $x>1$ ...
2
votes
3answers
81 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of R and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
1
vote
1answer
17 views

Proof from equality

suppose this equality holds: x, y, z are real 15(x + y + z) = 12(xy + yz + xz) = 10(x^2 + y^2 + z^2 ) and at least one variables isn't zero. I need to proof that x + y + z = 4 and find the ...
1
vote
2answers
37 views

First number in an open interval?

What is the first number in an open interval? For example, if I have the open interval (0, 1), what is the lowest number in that interval? Does this question even make sense with real numbers?
0
votes
1answer
22 views

How to prove these facts about integers using this definition?

A subset $A$ of $\mathbb{R}$, the set of real numbers, is said to be inductive if $1 \in A$ and if the statement $x \in A$ implies the statement $x+1 \in A$. The $Z_+$ of positive integers is ...
1
vote
2answers
44 views

Solving an unusual equation

I need to find a real number $n$ such that $n > 1$ and: $$ \sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n} $$ Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any ...
4
votes
1answer
147 views

Is the range of an injective function dense somewhere?

Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? ...
0
votes
2answers
34 views

Simplifying fractional exponents

I am very confused about the following: whenever I put in into wolfram alpha the answer it gives me is "indeterminate", is it not possible to simplify fractional exponents or something? if the ...
25
votes
9answers
3k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
0
votes
1answer
55 views

I think I was wrong: The supremum of a set whose elements squared less than a positive real number is square root of this number?

I think I was wrong... Can you find it out and teach me how to prove (the title)? If $c>0$ a real number. Define $E=\{x\in\mathbb R\mid x^2<c\}$. $E$ is nonempty because $0\in E$ (positivity ...
0
votes
1answer
28 views

Simplify the algebraic expression

Can someone please explain to me how the algebraic expression in the picture is simplified. To be more specific, how (1) becomes (2). $3x^2(6x-4)^4 + x^3(6\times 4\times (6x-4)^3)$ ...
1
vote
4answers
114 views

Prove or disprove the rationality of $ x^y $

Prove or disprove: "If $x$ is a rational number, and $y$ is an irrational number then $x^y$ is irrational" I am stuck with this, these are my steps. let $x=2$ and $y=\sqrt{2}$ ...
0
votes
2answers
41 views

Find a subset of the real numbers

I have to find an open and dense subset of the real numbers with arbitrarily small measure. Since the set of the rational numbers is dense, could we use a subset of the rationals?? How could I find ...
5
votes
4answers
91 views

Is $[a, a)$ equal to $\{a\}$ or $\varnothing$?

Let us define the set $[a,b) = \{ x \in \mathbb{R}: a\le x <b\}$ Is $[a, a)$ equal to $\{a\}$ or $\varnothing$?
1
vote
2answers
31 views

Rudin - Exercise 12, Cap. 2 Principles of Mathematical Analysis

Let $\Bbb{K}\subset\Bbb{R}$ consist of $0$ and the numbers $\frac{1}{n}$, for $n=1,2,3,\dots$. Prove that $\Bbb{K}$ is compact directly from the definition (without using the Heine-Borel theorem).
1
vote
1answer
31 views

For $f(x, y) = x-y$, is $f(K \times K)$ closed if $K$ is closed?

$f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y) = x-y$. For $K \subset \mathbb{R}$ closed is $f(K\times K)$ closed? For the closed interval this is straight forwardly true ...
-2
votes
1answer
50 views

Identify a countable union of nested intervals using the Archimedean principle [closed]

$\displaystyle\bigcup_{n=2}^\infty \left[\frac1n,3-\frac 2n\right]=(0,3)$. I can't prove using limits. I have to use the Archimedean principle and I don't know how to go about doing that..
2
votes
2answers
129 views

Is the function $\,f(x, y) = x-y\,$ closed?

Is the function $\,\,f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$, defined as $$f(x,y)=x-y,$$ closed?
4
votes
2answers
47 views

Show that if $a,b \in \Bbb R$ then [duplicate]

$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$ how would you go about solving this? I started with suppose $a \leq b$ Also, show min{a,b,c} = min{min{a,b},c}. How would ...
0
votes
0answers
27 views

Determine and sketch the pairs $(x,y)$ in $\mathbb{R} \times \mathbb{R}$ that satisfy some inequality

a) $|x| \leq |y|$ Continue my explanation below... If $y \geq 0$, then $-y \leq x \leq y$ and we get the region in the upper half-plane on or between the lines $y = x$ and $y = -x$
0
votes
1answer
90 views

Problem in Basic Real Analysis

I want to prove that if $$E_{b,y} = \{ r\in \Bbb Q |\; b^r\leq y \}$$ for $b\in \Bbb R^{>1}$ and $y\in \Bbb R^{>0}$ then, $\sup E_{b,y}$ is the unique solution to the equation $$b^x = y$$ I ...
2
votes
1answer
26 views

$A_\sqrt{2}$ isn't a Dedekind cut?

In a problem we consider a cut of $\mathbb{Q}$ a subset $A\subset\mathbb{Q}$ that fulfills: $A\neq\emptyset$ $\forall (q,q')\in A\times\mathbb{Q},\,q'<q\Rightarrow q'\in A$ $\forall q\in ...
2
votes
3answers
99 views

$\mathbf{Q}$ basis of $\mathbf{R}$.

Could someone give me an explicit basis of $\mathbf{R}$ as a vector space over $\mathbf{Q}$? I no some linearly independent subset, namely $1,e,e^2,\ldots$ but this seems to be a deep result already ...
0
votes
0answers
24 views

Checking this function for differentiability

$f(x) = |x|\sin x + |{x-\pi}|\cos x$ for $x \in \mathbb{R}$ Is the above function differentiable at $x=0$ ? At $x=\pi$ ?
-9
votes
7answers
301 views

How is greater than defined for real numbers?

I have an understanding of real numbers. For example, I can imagine real numbers as points on the line. The point which is more to the right represents bigger number. Or if I have decimal ...