All Questions
26
questions
11
votes
2
answers
978
views
Is there a mathematical statement that is linking integer limits to real limits?
I saw a question asking for the limit
$$\lim_{n \to \infty}\frac{\tan(n)}{n}.$$
At first I thought that the limit assumed $n$ to be a real number. So I gave the advice to use $\pi/2+2\pi k$ and $2\...
- 15.9k
9
votes
0
answers
138
views
If $n^x\in\Bbb Z,$ for every $n\in\Bbb Z^+,$ then $x\in\Bbb Z$ [duplicate]
Let $x$ is a real number such that $n^x\in\Bbb Z,$ for every positive integer $n.$ Prove that $x$ is an integer.
I got that problem here and it looks difficult, I tried writing $x$ as $\lfloor x\...
- 3,457
7
votes
2
answers
4k
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 $a \leq x < a+1$
...
- 2,157
6
votes
5
answers
5k
views
What is a natural number? [duplicate]
According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the ...
- 803
6
votes
2
answers
3k
views
can I find the closest rational to any given real, if I assume that the denominator is not larger than some fixed n
Given $n\in\mathbb{N}$ and $r\in\mathbb{R}$(or $r\in\mathbb{Q}$), is that possible to find a rational $\frac{a}{b}$such that, $b<n$ and $\frac{a}{b}$ is the "closest" rational to r?
With "closest"...
- 477
4
votes
6
answers
389
views
Why is $0^0$ undefined when $x^x=1$ as $x$ approaches $0$?
This question was born in another post available here. I believe $0^0=1$, because $x^x$ is continuous as $x$ approaches $0$.
Consider $\lim_{x \to 0}x^x$. Let $$f(x_n)=\bigg(\frac{1}{x}\bigg)^{\...
- 618
4
votes
2
answers
587
views
Cantor Sets in perfect sets in the Real numbers
My thesis is related with the Cantor sets. I was reading a lot of papers, blogs, etc, in order to look for the mean properties of these sets.
In one blog a read a proposition.
''Every perfect set ...
- 171
3
votes
1
answer
89
views
Irrationality of an "Euler-like" number
Let $(a_n)_{n=0}^{\infty}$ be a sequence of zeroes and ones such that $a_n=1$ for infinitely many $n$. Let $\displaystyle x:=\sum_{n=0}^{\infty} \frac{a_n}{n!} .$
Is $x$ irrational? I believe it is, ...
- 511
2
votes
1
answer
6k
views
Given a number ε > 0, prove there exists a natural number $N$ such that 1/N < ε
I believe there are three cases. I think I have figured out the first one.
Case 1: Let $ε$ = 1 and N > 1. Take $ N$=2. Then 1/2 < $ε$ ≡ 1/2 < 1.
Case 2: Let 0<$ε$<1. I believe that I ...
- 527
2
votes
4
answers
57
views
Bound a natural by two consecutive powers
I'm working in the following problems: Given two naturals $m$ and $n$, there exist a natural $d$ such that
$$m^{d}\leq n \leq m^{d+1}.$$
Afterwards I need to show that: If one chooses an arbitrary $...
- 1,167
2
votes
1
answer
234
views
were Irrational numbers discovered at Archimedes's age?
Archimedes axiom states a property of real numbers, while the real numbers include all the rational numbers and all the irrational numbers.
I wonder were Irrational numbers discovered at Archimedes's ...
- 2,267
1
vote
4
answers
1k
views
If $b-a>1$ then there is a $k\in \mathbb{Z}$ such that $a<k<b$
Given $a, b \in \mathbb{R}$, such that $b-a>1$, there is at least one $k\in \mathbb{Z}$ such that $a<k<b$.
My attempt:
Consider $E:=(a,b)\cap \mathbb{N}$. We need to show that $E$ is not ...
- 9,708
1
vote
2
answers
126
views
Please explain the proof of the validity of the Cauchy convergence criterion for real number sequences.
My question pertains to BBFSK, Vol I, Pages 143 and 144.
The following appears in the context of developing the real numbers as limits of sequences of rational numbers.
It is also easy to prove ...
- 3,349
1
vote
1
answer
49
views
Can we find a relation between the three integrers $m$, $j$ and $k$?
Let $r>4$ and $n>1$ positive integers and let $α$ be a positive real number.
Let us define the following three positive integers:
$$
\begin{align*}
m &= \lfloor r^{(n+1)^2} \alpha \rfloor \...
- 3,021
1
vote
2
answers
113
views
Proving any $x \in [0,1]$ belongs to infinitely many $S^{k}_{n}:=[\frac{k-1}{2^n},\frac{k}{2^n}]$
I am trying to prove the following:
Define $S^{k}_{n}:=[\frac{k-1}{2^n},\frac{k}{2^n}]$
i) Given any $x \in [0,1]$, then $x$ belongs to infinitely many $S^{k}_{n}$
ii) Any $x \in [0,1]$ also belongs ...
- 293