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Questions tagged [real-numbers]

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

1
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1answer
18 views

Suppose $x$ is a non-negative real number such that for all $\epsilon >0$ we have $x < \epsilon$. Then $x=0$

Is the following statement true? Suppose $x$ is a nonnegative real number such that for all $\epsilon >0$ we have $x < \epsilon$. Then $x=0$. Intuitively speaking yes since $0$ is an infimum ...
2
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1answer
29 views

I don't understand why this axiom makes the difference between a completely ordered field and an ordered field

I've been reading about ordered fields and completely ordered fields, and I'm stuck on the difference between the rational and real numbers that makes the rational numbers an ordered field and the ...
4
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1answer
46 views

Could Someone Explain an Ordered Field in Layman's Terms?

High school student here... I am currently working on Spivak and I'm not very far (just reached questions in chapter 1). I was discussing my progress with someone recently and they asked me to ...
0
votes
2answers
25 views

Convergence test for a series

Let S = $\sum_{n=1}^{\infty}{\frac{n^k}{((n^3+n)^{\frac{1}{3}}-n)}},\forall\space\space\space\space k\in\mathbb{Z}.$ Say if this series converge of diverge. My attempt: $$S=\sum_{n=1}^{\infty}\frac{...
2
votes
3answers
42 views

Homeomorphism between $\mathbb{R}$ and $(0,1)$ [duplicate]

I was working with an exercise of general topology and I had a question: are there an homeomorphism $f:\mathbb{R}\to(0,1)$ such that $f(x)\in\mathbb{Q}$ if and only if $x\in\mathbb{Q}$?, i.e., the ...
0
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2answers
47 views

Prove $\frac1{\sqrt{(x_n)}}$ converges to $\frac1{\sqrt{(x)}}$

So far I have: $$\vert\frac1{\sqrt{(x_n)}} - \frac1{\sqrt{(x)}}\vert = \vert\frac{\sqrt{(x)}-\sqrt{(x_n)}}{\sqrt{(x)}\sqrt{(x_n)}}\vert = \frac{\sqrt{(x)} + \sqrt{(x_n)}}{\sqrt{(x)}\sqrt{(x_n)}}$$ ...
0
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0answers
31 views

Are there more real numbers than irrational numbers? [duplicate]

Wrapping my head around the mathematical definition of infinity and just curious here: Are there more real numbers than irrational numbers? It would intuitively seem so, but they are both just ...
3
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0answers
15 views

Can the constant $3$ in the Vitali covering lemma be replaced by any positive constant less than that in the finite case?

I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case. Observe that this question is different ...
0
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1answer
75 views

Where is my mistake in this integral?

$$ \int_0^\infty \frac{t}{e^t} dt $$ The above integral can be computed to be exactly 1 with integration by parts. However, i just wanted to try different integration techniques to see if i could ...
-1
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2answers
29 views

Why does a uniformly continuous function on [a,b] in the reals need be closed AND bounded?

I cannot come up with any counter examples as to why a function would need to be both closed and bounded to be uniformly continuous. Why is it not sufficient to just have one condition? For example, ...
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1answer
10 views

Can we say for any given interval in $\mathbb R$ that every point in this interval is accumulation point?

Accumulation point means briefly that if $x_0$ is acc. point in set $S$ then for any given $\epsilon>0$, $((x_0-\epsilon, x_0+\epsilon ) \cap S )\setminus\{x_0\} \not = \emptyset$ My reasoning is,...
1
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1answer
31 views

Suppose that $b\ge f(x) \ge a$ for all $x\in \Bbb{R}$ with $x \neq c$, and suppose $\lim_{x\to c}f(x)=L$. Prove $b \ge L \ge a$?

Let $f:\Bbb{R}\to\Bbb{R}$. Suppose that $b\ge f(x) \ge a$ for all $x\in \Bbb{R}$ with $x \neq c$, and suppose $\lim_{x\to c}f(x)=L$. Prove $b \ge L \ge a$. How do I go about tackling this problem. If ...
0
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1answer
48 views

Where is my mistake…? [duplicate]

Let $$x=0.99999\ldots.$$ Clearly $x$ is a rational number. I want to find $a,b$ such that $$x=\frac{a}{b}.$$ Clearly $10x-x=9$ and thus $x=1$. So $$1=0.99999\ldots$$ Where is my mistake?
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0answers
36 views

Find all pairs x,y belong to Z [closed]

Find all integral solutions (positive or negative) of $x$ and $y$ so that $$x^3 + x^2 y + xy^2 + y^3 = 8(x^2+ xy + y^2 + 1)$$
1
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2answers
17 views

Interpolation Problem?

I'm making a computer program and I ran into a problem of mapping/interpolating number from one range into another. I've formalized it as a mathematics problem. Look over it and I'd appreciate your ...
0
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1answer
28 views

Problem in proof of -“A net has $y$ as a cluster point iff it has a subnet which converges to $y$”

Directed Set: We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$ such that (i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$; (ii) x ≤ x for each x ∈ $\omega$; (iii) ...
0
votes
0answers
13 views

When does a set of inequalities over series imply an inequality over variables?

I asked a preliminary question here. This is a followup question. Let $X=(x_1,...x_n)$ be a set of variables with $x_i\in \mathbb R$, for each $i\in I$ where $I=\{1,..,n\}$ is the indicator set (of ...
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2answers
27 views

Does every real number have a unique binary representation?

I'm actually intereseted in real numbers that belong to the interval $(0,1)$, but a more general answer will be great
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3answers
31 views

Inequality in difference of square roots

I think this inequality is true and I'm trying to prove it: For real, non-negative $a$ and $b$ $$\lvert \sqrt{a} - \sqrt{b}\rvert \le \lvert \sqrt{a - b} \rvert$$ Closest thing I've found is ...
0
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1answer
30 views

If $a-b\le c<\infty$ then $a<\infty$

Given arbitrary $a,b$ in the extended reals do we really know that; If $a-b\le c<\infty$ then $a<\infty$
0
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1answer
41 views

If $e^{f(x)}$ is continuous, does it follow that $f$ is continuous?

If $e^{f(x)}$ is continuous, does it follow that $f$ is continuous? I say yes. Proof by contradiction Suppose not. That is, suppose $e^{f(x)}$ is continuous and $f$ is not continuous. Then $$\...
1
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1answer
35 views

If f and g are functions such that f + g is continuous, does it follow that at least one of f or g must be continuous?

If $f$ and $g$ are functions such that $f + g$ is continuous, does it follow that at least one of $f$ or $g$ must be continuous? I said yes. Proof by contradiction. Suppose not, that is, suppose $f$ ...
10
votes
2answers
2k views

How can I mathematically split up a 3 digit number?

For example, if I have 456, How can I split this and then let each column value be a separate number? The only way I can think of doing this would be to subtract '100' n times until that column is '0' ...
0
votes
2answers
28 views

Why given algebra three numbers are of the same sign?

Let $x,y,z$ be given real numbers. Why if $xyz(x+y+z)>0$ and $xy+xz+yz>0$, then $x,y,z$ are of the same sign?
1
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1answer
55 views

Can inequalities over $n>2$ variables ever imply an inequality over $2$ variables?

Say we have $n$ variables $x_1,x_2...,x_n$. Question 1. We want to put conditions of the following form: $$x_i+...+x_j>x_k+...+x_l$$ Where The left side contains as many variables as the right ...
1
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1answer
33 views

Why this property of open sets in $\mathbb{R}^1$ cannot be generalized on the large dimensions

In textbook I found the following statement: "Any open set on a line is a disjoint union of finite or countable number of intervals." I undertand the proof (but cannot understand, why the disjoint ...
0
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0answers
39 views

Bounds on real numbers

Suppose that $\{\epsilon _1 , · · · , \epsilon_m \} $are real numbers that all satisfy $|\epsilon_i | ≤ \eta$. Show that given $C > 1$, we have that $$ \Pi_{j=1}^m(1+\epsilon_j) = 1 + \epsilon$$...
2
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2answers
30 views

Represent the interval [0,1) by union

I can represent the set $(0,1]$ like: $(0,1] = \bigcap_{n=1}^{\infty} (0, \frac{n+1}{n}) = \bigcup_{n=1}^{\infty} [\frac{1}{n},1]$. But how can I represent $[0,1)$? I have a task, which requires ...
3
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0answers
30 views

An introduction book for analysis with the axioms of Tarski

Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis? (I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
2
votes
1answer
22 views

Cartesian product of countable number of finite sets is countable.

Cartesian product of countable number of finite sets is countable. True or false? I think this statement is false. Because consider $A = \{1,0\}$ . If we compute countable product of $A$ with itself, ...
0
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0answers
22 views

Proving closed interval [0,1] has same cardinality as Real Numbers [duplicate]

I want to prove that the closed interval [0, 1] has same cardinality as Real Numbers. I was able to figure out (0, 1), but need help with proving the closed part. What I have so far is: the function $...
0
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0answers
30 views

Excercise on meet and join on real numbers

The statement Given that for $ \alpha,\beta\in(\mathbb{R},{\leqq}) $, $ \alpha\wedge\beta = \min\{\alpha,\beta\} $ and $ \alpha\vee\beta = \max\{\alpha,\beta\} $, let $a_1,a_2,b_1,b_2$ be real ...
0
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2answers
33 views

Absolute Value Proof By Cases

I'm currently working through D. Velleman's How to Prove it. I have a question regarding an absolute value proof by cases (#10; section 3.5). The question asked is to prove that: $$ \forall x\in\...
0
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2answers
28 views

Transform a totally ordered set to a structure that is isomorphic to (R,+,.,≤)

So let $(M,\le_M)$ be a totally ordered set. Can we define $+$ and $.$ to make $M$ isomorphic to $(\mathbb{R},+,.,\le)$? I mean the well known axioms. To let this possible: $M$ is not bounded above ...
1
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3answers
32 views

Show the equivalence $x=1$ $⟺$ $-ε<x-1<ε$ for all $ε>0$

Let $x\in \mathbb{R}$. Show the equivalence $x=1 \Leftrightarrow-\epsilon\lt (x-1)\lt \epsilon$ for all $ε\gt0$. So the first thing I thought to do was to prove both sides ($\Leftarrow$ and $\...
0
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1answer
23 views

Find real numbers satisfying some conditions of rational dependance

Does there exist $\xi_1,\ldots,\xi_4\in\mathbb R$ such that $$\forall (\alpha,\beta)\in\mathbb Q^2\setminus\{0,0\},\quad \begin{cases}\dim_{\mathbb Q}(\alpha \xi_1+\beta\xi_3,\,\alpha\xi_2+\beta\...
0
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0answers
23 views

Proof for Floor function exponent algebra Identity

With the exeption of $x$, The end goal for me here is to determine the maximum cardinality subset of the reals over three variables ${\{(r,y,z) \in \mathbb R^3}\}$ for which the equality below holds, ...
0
votes
1answer
52 views

Assume $(a_n)$ and $(b_n)$ are sequences in X and let $x \in X$ and that $(a_n) \to x$ and $d(a_n,b_n) \to 0$. Prove that $(b_n) \to x$. [closed]

Let (X,d) be a metric space. Assume that (a_n) and (b_n) are sequences in X and let x be an element of X. Assume also that a_n converges to x and d(a_n,b_n) converges to 0. Prove that (b_n) converges ...
0
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3answers
54 views

Is this statement valid for positive $x$ and $y$? If $x>y$, then $\frac{1}{x} < \frac{1}{y}$.

Is this following statement valid (where both $x$ and $y$ are positive)? If $x>y$, then $\dfrac{1}{x} < \dfrac{1}{y}$.
2
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4answers
158 views

Is this proof of the totality of this relation correct?

I'm currently writing an original chapter of a textbook I'm republishing and I need to know if this rather pedantic proof of a total ordering on $\mathbb{R}$ is correct. It looks correct, but ...
0
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0answers
13 views

Is comparing real and complex values within Robin's Inequality legal? And how would we?

I have a problem where I need to compare real and complex numbers. I see here and here that there are different ways to go about interpreting the sizes of complex numbers, but in my context I want to ...
0
votes
1answer
38 views

The properties of real numbers field [closed]

I know, that the multiplicative group of $\mathbb{R}$ is create on the set $\mathbb{R}\setminus \{0\}$. But how we can multiply real numbers on the $0$ after this? This point was unswered, I think. ...
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1answer
68 views

Is the inequality $(\sqrt{x} - \sqrt{y})^2 \geqslant 0$ always true for $x,y \in \mathbb{R}^+$?

Suppose $x,y \in \mathbb{R}^+$ Then is the following inequality true? $(\sqrt{x} - \sqrt{y})^2 \geqslant 0$ If it is not true then please provide an example of why it fails.
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0answers
11 views

Generating a grid that scales nicely

Let two positive values $x > y$ be given along with an integer $h$. Consider the interval $[0, x]$. I want to split this up into $h$ subintervals each of equal length. This is of course always ...
0
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1answer
38 views

Prove that $\mathbb{R^n}$ is not union of a finite number of closed sets with empty interior

I don't know how to finish this one. Here's what I've done so far: We're trying to find a contradiction. Suppose that $\mathbb{R^n}=C_1\cup C_2 \cup \dots \cup C_n$, with $C_i$ closed subsets ...
2
votes
0answers
36 views

Why do products simplify?

I’m trying to put in more rigorous math terms what it means to simplify a product of any two abstract things (in the sense that I mean it), and what makes it work. Here’s an example: $2\times3=6$, ...
2
votes
3answers
88 views

If $a>1$. Prove that, for any real numbers $x, y:a^{x+y} = a^x a^y$

If $a>1$. Prove that, for any real numbers $x, y$:$$a^{x+y} = a^x a^y$$ $$(a^x)^y = a^{xy}$$ I feel that if $x,y$ were natural numbers the proof could be by induction but in our case how it can ...
1
vote
2answers
34 views

Prove that $n$ is divisible by $5$ if and only if $n^2 $is divisible by $5$ and use it to prove that $\sqrt{5}$ is irrational

Not sure about what cases to consider in part 1 of the proof and how to use it to prove the next part.
0
votes
3answers
39 views

Proof using axioms of real numbers [closed]

Using axioms of real numbers prove for ∀x∈R, ∀y∈R,∀n∈N: ($0 \leq x < y$ ) $\Rightarrow$ $x^n < y^n$ Any help with this? I have no idea how to even begin.
0
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0answers
60 views

Prove some linear algebra equalities and how to geometric interpret them

Let $\alpha,\beta \in\mathbb{R}$ such that $\alpha + \beta = 1$ and $x,y,z \in V$ where $V$ is a vector space. Show that $$\|z-(\alpha x+\beta y)\|^2=\alpha\|z-x\|^2+\beta\|z-y\|^2-\alpha\beta \|...