Questions tagged [real-numbers]
For questions about $\mathbb{R}$, the field of real numbers. Often used in conjunction with the real-analysis tag.
1
vote
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
votes
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
votes
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:
$$S=\sum_{n=1}^{\infty}\frac{...
2
votes
3answers
42 views
Homeomorphism between $\mathbb{R}$ and $(0,1)$ [duplicate]
I was working with an exercise of general topology and I had a question: are there an homeomorphism $f:\mathbb{R}\to(0,1)$ such that $f(x)\in\mathbb{Q}$ if and only if $x\in\mathbb{Q}$?, i.e., the ...
0
votes
2answers
47 views
Prove $\frac1{\sqrt{(x_n)}}$ converges to $\frac1{\sqrt{(x)}}$
So far I have:
$$\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)}}$$
...
0
votes
0answers
31 views
Are there more real numbers than irrational numbers? [duplicate]
Wrapping my head around the mathematical definition of infinity and just curious here: Are there more real numbers than irrational numbers? It would intuitively seem so, but they are both just ...
3
votes
0answers
15 views
Can the constant $3$ in the Vitali covering lemma be replaced by any positive constant less than that in the finite case?
I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different ...
0
votes
1answer
75 views
Where is my mistake in this integral?
$$ \int_0^\infty \frac{t}{e^t} dt $$
The above integral can be computed to be exactly 1 with integration by parts.
However, i just wanted to try different integration techniques to see if i could ...
-1
votes
2answers
29 views
Why does a uniformly continuous function on [a,b] in the reals need be closed AND bounded?
I cannot come up with any counter examples as to why a function would need to be both closed and bounded to be uniformly continuous. Why is it not sufficient to just have one condition?
For example, ...
-1
votes
1answer
10 views
Can we say for any given interval in $\mathbb R$ that every point in this interval is accumulation point?
Accumulation point means briefly that if $x_0$ is acc. point in set $S$ then for any given $\epsilon>0$, $((x_0-\epsilon, x_0+\epsilon ) \cap S )\setminus\{x_0\} \not = \emptyset$
My reasoning is,...
1
vote
1answer
31 views
Suppose that $b\ge f(x) \ge a$ for all $x\in \Bbb{R}$ with $x \neq c$, and suppose $\lim_{x\to c}f(x)=L$. Prove $b \ge L \ge a$?
Let $f:\Bbb{R}\to\Bbb{R}$. Suppose that $b\ge f(x) \ge a$ for all $x\in \Bbb{R}$ with $x \neq c$, and suppose $\lim_{x\to c}f(x)=L$. Prove $b \ge L \ge a$.
How do I go about tackling this problem. If ...
0
votes
1answer
48 views
Where is my mistake…? [duplicate]
Let
$$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$$
Where is my mistake?
-1
votes
0answers
36 views
Find all pairs x,y belong to Z [closed]
Find all integral solutions (positive or negative) of $x$ and $y$ so that
$$x^3 + x^2 y + xy^2 + y^3 = 8(x^2+ xy + y^2 + 1)$$
1
vote
2answers
17 views
Interpolation Problem?
I'm making a computer program and I ran into a problem of mapping/interpolating number from one range into another. I've formalized it as a mathematics problem. Look over it and I'd appreciate your ...
0
votes
1answer
28 views
Problem in proof of -“A net has $y$ as a cluster point iff it has a subnet which converges to $y$”
Directed Set:
We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$
such that
(i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$;
(ii) x ≤ x for each x ∈ $\omega$;
(iii) ...
0
votes
0answers
13 views
When does a set of inequalities over series imply an inequality over variables?
I asked a preliminary question here. This is a followup question.
Let $X=(x_1,...x_n)$ be a set of variables with $x_i\in \mathbb R$, for each $i\in I$ where $I=\{1,..,n\}$ is the indicator set (of ...
0
votes
2answers
27 views
Does every real number have a unique binary representation?
I'm actually intereseted in real numbers that belong to the interval $(0,1)$, but a more general answer will be great
0
votes
3answers
31 views
Inequality in difference of square roots
I think this inequality is true and I'm trying to prove it:
For real, non-negative $a$ and $b$
$$\lvert \sqrt{a} - \sqrt{b}\rvert \le \lvert \sqrt{a - b} \rvert$$
Closest thing I've found is ...
0
votes
1answer
30 views
If $a-b\le c<\infty$ then $a<\infty$
Given arbitrary $a,b$ in the extended reals do we really know that;
If $a-b\le c<\infty$ then $a<\infty$
0
votes
1answer
41 views
If $e^{f(x)}$ is continuous, does it follow that $f$ is continuous?
If $e^{f(x)}$ is continuous, does it follow that $f$ is continuous?
I say yes.
Proof by contradiction Suppose not. That is, suppose $e^{f(x)}$ is continuous and $f$ is not continuous. Then
$$\...
1
vote
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
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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:
$$
\forall x\in\...
0
votes
2answers
28 views
Transform a totally ordered set to a structure that is isomorphic to (R,+,.,≤)
So let $(M,\le_M)$ be a totally ordered set.
Can we define $+$ and $.$ to make $M$ isomorphic to $(\mathbb{R},+,.,\le)$?
I mean the well known axioms.
To let this possible:
$M$ is not bounded above ...
1
vote
3answers
32 views
Show the equivalence $x=1$ $⟺$ $-ε<x-1<ε$ for all $ε>0$
Let $x\in \mathbb{R}$. Show the equivalence $x=1 \Leftrightarrow-\epsilon\lt (x-1)\lt \epsilon$ for all $ε\gt0$.
So the first thing I thought to do was to prove both sides ($\Leftarrow$ and $\...
0
votes
1answer
23 views
Find real numbers satisfying some conditions of rational dependance
Does there exist $\xi_1,\ldots,\xi_4\in\mathbb R$ such that
$$\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
votes
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
votes
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
votes
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
votes
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. ...
-1
votes
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.
0
votes
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
votes
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
votes
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 \|...
