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

Why are positive rational numbers countable but real numbers are not?

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
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2answers
28 views

Is Z3 a sub-field of R?

The inverse numbers for the items in $\mathbb{Z3}$ are different than in $\mathbb{R}$ so I assume it's not a sub-field of $\mathbb{R}$. Am I correct? And in general, can sub-field of a infinite ...
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2answers
52 views

Verification of this proof that the set of real numbers is uncountable.

I don't want a proof. I just want verification or correction of the proof I supplied: We start with the fact that between any two real numbers there is a rational number. There is an infinite ...
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1answer
88 views

Does every positive integer appear in the digits of $2\cdot 0.1234567891011… $?

Let $C = 0.1234567891011121314…$ the Champernowne constant. My question is : Does the real number $2 \cdot C \simeq 0.24691357820222426283032343638404244464850525456586062646668707274...$ contain ...
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1answer
14 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
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0answers
24 views

Prove that If $A \subseteq \Bbb{R}$ non-empty lower bounded then $\exists$ the infimum of A

I am trying to prove this proposition proof Suppose that $A \subseteq \Bbb{R}$ non-empty lower bounded, that is $\forall x \in A$ $x \ge \alpha$ for some $\alpha \in \Bbb{R}$ case 1 $\alpha$ is the ...
2
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2answers
48 views

If $\{a_n\}$ and $\{b_n\}$ are increasing, then $\{a_n b_n\}$ is increasing.

For two sequences $a_n$ and $b_n$, “If $\{a_n\}$ and $\{b_n\}$ are increasing, then $\{a_nb_n\}$ is increasing.” Show this is false, make the hypothesis on $\{b_n\}$ stronger, and prove the amended ...
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1answer
36 views

Prove that $||a|-|b||\leq |a-b|$ for all real numbers

Prove that $||a|-|b||\leq |a-b|$ for all real numbers I was thinking divide it into $a\geq b$ and $a<b$, but then I realized I need to include situations when they are greater than zero and less ...
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1answer
19 views

if $-a\leq b\leq a$, then $|b|\leq a$

if $-a\leq b\leq a$, then $|b|\leq a$ I started by showing that $-|b|\leq b\leq |b|$, but then I can only see that $a$ can be equal to $|b|$, how do I show $|b|\leq a$?
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1answer
24 views

Proof of $a \le 0$ $\Leftarrow \Rightarrow$ $\forall \epsilon > 0$ $a < \epsilon$

In analysis class I saw a proof but I would like see another Proof: $\Rightarrow$ Suppose that $a \le 0$ $\land$ $\epsilon > 0$ if $a=0$ by hypothesis $a < \epsilon$ if $a<0$ $\Rightarrow$ ...
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2answers
68 views

proof of 1) $|a| = 0 \iff a = 0$ and 2) $|a| \ge 0$

On the internet there are many proofs but very summary Definition $|a| = $ $a$ if $a \ge 0$ $-a$ if $a < 0$ Proposition 1 $|a| = 0 \Leftarrow \Rightarrow a = 0$ $\Rightarrow$) Suppose that $|a| ...
1
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1answer
31 views

$A$ and $B$ be non-empty bounded set of real numbers, give a counter example to the following.

Assume $A \cap B \neq \emptyset$. Find a counter-example to the claim: $\sup(A \cap B) = \min\{\sup(A), \:\sup(B)\}$ I cant seem to find a counter example to the above claim, can anyone provide a ...
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3answers
47 views

If $b-a>1$ then there is a $k\in \mathbb{Z}$ such that $a<k<b$

Given $a, b \in \mathbb{R}$, such that $b-a>1$, there is at least one $k\in \mathbb{Z}$ such that $a<k<b$. My attempt: Consider $E:=(a,b)\cap \mathbb{N}$. We need to show that $E$ is not ...
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0answers
25 views

Well-ordering principle proof via Analysis

I would appreciate if my proof attempt could be evaluated, and some hints could be given. I think that, perhaps, my proof is not ideal. Prove: If $E$ is a non-empty subset of $\mathbb{N}$ then $E$ ...
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0answers
18 views

Proof of Well-ordering principle [closed]

In analysis class we are building natural numbers from the real numbers The teacher gave us a Theorem Theorem(Well-ordering principle) Let $G \subseteq \Bbb{N}$ non-empty then $\exists m_0$$\in G: ...
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1answer
31 views

$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$

I'd appreciate if you could please check to see if my proof is valid. Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$. Solution: $q^2 > p^2/5 \iff q > ...
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3answers
87 views

Is $(-\infty, 0)$ the same size as $(0, \infty)$?

A differential equations problem asked about the largest interval on which the solution was defined. The solution was defined except for $t=0$, which made me wonder whether the intervals $(-\infty, ...
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0answers
49 views

How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the ...
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2answers
80 views

Is the element $(0,0,0)\in\mathbb{R}^3$ a divisor of zero?

(I'm assuming that $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$) In my assignment, I'm told to prove that exactly one of the following can be true for an element $(x,y,z)\in\mathbb{R}^3$ ...
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1answer
37 views

two positive real numbers have their sum, product… [closed]

Two positive real numbers have their sum,product and difference of the squares $(a^2-b^2)$ equal. Find those numbers. It would be easy to solve if only two of these were mentioned, but I don't know ...
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0answers
19 views

Comparing irrational powers

I want to compare two numbers of the form $a^b$ and $b^a$ for $a,b, >1$. The simpler way that I know is to start from: $$ a^b\le b^a \quad \iff \quad b\log a\le a \log b \quad \iff \quad ...
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0answers
32 views

Proof of $\Bbb{N}$ is inductive

Here there is a proof but I think it is incomplete(missing $1\in\Bbb{N}$) Note: A⊆R is inductive if and only if 1∈A and ∀x∈A⇒x+1∈A. By definition, $\Bbb{N}$ is the intersection of all inductive sets ...
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0answers
38 views

Proof of If $A \subseteq \Bbb{R}$ with $A$ inductive set then $\Bbb{N}\subseteq A$

In calculus class, I saw a proof of this, but I am not convinced. Note: A$\subseteq \Bbb{R}$ is inductive if and only if 1$\in A$ and $\forall x \in A \Rightarrow x+1 \in A$. I tried to do a proof ...
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2answers
47 views

Use the Intermediate Value Theorem to prove that $\sqrt s$ exists?

I need to prove with the Intermediate Value Theorem that $\sqrt s$ exists, where $s > 0$. My textbook states this definition of the Intermediate Value Theorem: Suppose that $f$ is continuous on ...
0
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1answer
23 views

ideal number gernerator

I was trying to solve a problem on Hackerearth. Here: https://www.hackerearth.com/problem/algorithm/ideal-random-number-generator/ I solved this partially:https://ideone.com/pXkHwQ (passed three ...
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3answers
60 views

Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$. [duplicate]

How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset ...
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1answer
28 views

Proof of $\forall a \in \Bbb{R} -a = (-1)a$

Problem. Prove that $ \forall a \in \Bbb{R}: -a = (-1) a $. The teacher gave us a proof but I would like to see another :) Proof by contradiction Suppose that $ -a \neq (-1) a $. Then ...
2
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2answers
53 views

Prove that $\times : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous

I'm trying to solve the first batch of exercises from Topology and Groupoids, but I got stuck with the following one: Let $C$ be a neighborhood of $c \in \mathbb{R}$, and let $ab = c$. Prove that ...
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1answer
34 views

Give an example of two closed disjoint sets $F$ and $G$ (subsets of $\mathbb{R}$) such that $\inf\{|x-y|; x\in F, y\in G\}=0$.

Give an example of two closed disjoint sets $F, G\subset\mathbb{R}$ such that $\inf\{|x-y|; x\in F, y\in G\}=0$. I tried so much! I found that both have to be unbounded, because if one of them is ...
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3answers
40 views

What is the `e` notation regarding to decimal numbers?

Say the following number: $\text{1.4E-46}$. What is the e meaning? I'm not talking about euler number. Thanks in advance.
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1answer
33 views

When is $\cap_{i=1}^{\infty} A_i$ non-empty

$A_1\supset A_2\supset\cdots A_n\supset A_{n+1}\supset\cdots$ be an infinite sequence of non-empty subsets of $\mathbb R^3$.Which one of the following ensures that their intersection ...
3
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1answer
90 views

Let $x$ be a real number. Prove the existence of a unique integer $a$ such that $a \leq x < a+1$

Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$ ...
0
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1answer
27 views

Proof of $ \forall a \in \Bbb{R}: -a = (-1) a $.

Problem. Prove that $ \forall a \in \Bbb{R}: -a = (-1) a $. I already have a proof but I would like to see another. :) Proof by contradiction Suppose that $ -a \neq (-1) a $. Then ...
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1answer
66 views

Defining $\mathbb{R}$ as a field

One of the definitions of real numbers I've encountered is "Dedekind complete totally ordered field". How to prove said field is unique? The definition seems to be not complete too, since rational ...
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2answers
31 views

Under what conditions is $\sup(A)$ not an accumulation point for $A$? $A$ is a subset of $\mathbb{R}$

I am having a hard time conceptualizing this question. I can't see any conditions that would make it soI couldn't find an accumulation point. If $m = \sup(A)$, then can't we always find some ...
3
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1answer
79 views

Is there a choice homomorphism?

Let $\pi : \mathbb{R} \to \mathbb{R}/ \mathbb{Q}$ be the canonical projection. With the axiom of choice we "know" that there are choice functions $\alpha : \mathbb{R}/ \mathbb{Q} \to \mathbb{R}$ with ...
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2answers
27 views

Find the real and imaginary parts of the following.

$$\frac{z-a}{z+a}; a \in \Re$$ The part I'm confused about is the $a \in \Re$. I know that this means that $a$ is a real number (not imaginary), but then how do I interpret the addition/subtraction ...
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4answers
1k views

Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is ...
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0answers
17 views

number formed by inserting odd number of zeros between two ones

we know that $101$ is a prime number. similarly $10001$ is also prime. Can we have a general proof that number with trailing ones and odd number of zeros between them is Prime. That is if ...
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2answers
20 views

Finding range of $(a+d)(b+c)$

Four positive real nos. satisfy the equation: $a+b+c+d = 2$ Let $M = (a+c)(d+b).$ Find the range of $M.$ I try for values, and find that minimum cannot be less than or equal to 0. The maximum ...
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1answer
44 views

Step 6 Real construction from Rational Baby Rudin

Rudin's Books establishes Multiplication is a little more bothersome than addition in the present context , since products of negative rationals are positive. For this reason we confine ...
102
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13answers
14k views

Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
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0answers
57 views

I can't understand the formal definition of $\mathbb{R}$

I've always intuitively understood this set in intuitive sense, as "all numbers on the number line". However, now I want to know the formal definition: Consider the set of rational numbers, ...
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1answer
22 views

Non-zero Number Multiplication

How would you solve this problem: If $a,b,c$ are non-zero real numbers such that $\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$, and $x=\frac{(a+b)(b+c)(c+a)}{abc}$, and $x<0$, then $x$ equals ...
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1answer
33 views

Is there an irrational number arbitrarily close to another irrational number?

I know that there is a rational number arbitrarily close to an irrational, due to the density of real number. But what about an irrational number? Thanks!
3
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1answer
35 views

Can I extend mathematical induction to real numbers? [duplicate]

Here is my rather simple idea. I will treat the set of real numbers as a set of discrete continuities, each separated by an Epsilon ball that tends to 0. So, let's say P(b) is true. We then assume ...
2
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2answers
34 views

Topology on $\mathbb{R}$ with certain property

I am trying to find a topology on $\mathbb{R}$ that is not discrete or indiscrete, where every open set is closed. My idea was the topology $T$ generated by $\{ (k,k+1):\;k\in \mathbb{Z}\}\cup \{ ...
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1answer
51 views

A property of the Cantor set. [closed]

Let $C$ be the Cantor set. Prove $C-C=[0,1]$ where $A-B$ is defined to be $\{x-y\,:\, x\in A,\,y\in B\}$. We can't use the ternary expansion.
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3answers
40 views

Disprove bijection between reals and naturals

Coming across diagonalization, I was thinking of other methods to disprove the existence of a bijection between reals and naturals. Can any method that shows that a completely new number is created ...
2
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0answers
28 views

Clarification on the definition of Supremum

Most sources that define an upper bound as: If $A$ is a set of numbers and $b$ is a number, then $b$ is an upper bound if and only if $x \le b $ for all $x \in A$ I have three questions: Since ...