Set of Natural Numbers
Let, $A$ be a non-empty set of natural numbers.
And denote, $\{n\} = |\{a | a \in A, a \leq n\} |$ and $[ n ] = \frac{ \{n\} }{n} $for all natural numbers $n$
Given that, $[m] \geq [n]$ for all $m \geq n$
Prove ( or disprove ) that, for any $a,b \in \mathbb{N}$
\[ \lim_{n\to\infty}\frac{ \{ n \} }{ \{ a n + b \} } = \frac{1}{a}\]
And denote, $\{n\} = |\{a | a \in A, a \leq n\} |$ and $[ n ] = \frac{ \{n\} }{n} $for all natural numbers $n$
Given that, $[m] \geq [n]$ for all $m \geq n$
Prove ( or disprove ) that, for any $a,b \in \mathbb{N}$
\[ \lim_{n\to\infty}\frac{ \{ n \} }{ \{ a n + b \} } = \frac{1}{a}\]
ধনঞ্জয় বিশ্বাস
Re: Set of Natural Numbers
Hints:
Is $A$ finite?
What does the well ordered property of $\mathbb {N}$ imply?
Solution:
Is $A$ finite?
What does the well ordered property of $\mathbb {N}$ imply?
Solution:
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor
Re: Set of Natural Numbers
A slightly modified ( and may be harder ) version:
Let, $A$ be a non-empty set of natural numbers.
And denote, $\{n\} = |\{a | a \in A, a \leq n\} |$ and $[ n ] = \frac{ \{n\} }{n} $for all natural numbers $n$
Given that, $[a n + b] \geq [n]$ for all $n\in \mathbb{N}$
Prove ( or disprove ) that,
\[ \lim_{n\to\infty}\frac{ \{ n \} }{ \{ a n + b \} } = \frac{1}{a}\]
Let, $A$ be a non-empty set of natural numbers.
And denote, $\{n\} = |\{a | a \in A, a \leq n\} |$ and $[ n ] = \frac{ \{n\} }{n} $for all natural numbers $n$
Given that, $[a n + b] \geq [n]$ for all $n\in \mathbb{N}$
Prove ( or disprove ) that,
\[ \lim_{n\to\infty}\frac{ \{ n \} }{ \{ a n + b \} } = \frac{1}{a}\]
ধনঞ্জয় বিশ্বাস
Re: Set of Natural Numbers
???Corei13 wrote:A slightly modified ( and may be harder ) version:
Let, $A$ be a non-empty set of natural numbers.
And denote, $\{n\} = |\{a | a \in A, a \leq n\} |$ and $[ n ] = \frac{ \{n\} }{n} $for all natural numbers $n$
Given that, $[a n + b] \geq [n]$ for all $n\in \mathbb{N}$
Prove ( or disprove ) that,
\[ \lim_{n\to\infty}\frac{ \{ n \} }{ \{ a n + b \} } = \frac{1}{a}\]
$[a n + b] \geq [n]$ or $\frac {\{an+b\}} {an+b} \geq \frac {\{n\}} {n}$
Or, $\frac {\{n\}} {\{an+b\}} \leq \frac n {an+b}$
Now, I'm not sure about this part,
In the limiting case $ \lim_ {n\to\infty}$ $\frac {\{n\}} {\{an+b\}} = \lim_ {n\to\infty} \frac n {an+b} = \frac 1 a$
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: Set of Natural Numbers
From $\frac {\{n\}} {\{an+b\}} \leq \frac n {an+b}$, we have $\lim_ {n\to\infty} \frac {\{n\}} {\{an+b\}} \leq \frac {1}{a}$, we don't have the equality.
ধনঞ্জয় বিশ্বাস
Re: Set of Natural Numbers
Shouldn't my earlier proof work in this case also?
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor
Re: Set of Natural Numbers
In our earlier proof we got an equality, but here I'm getting an inequality as Mahi's
ধনঞ্জয় বিশ্বাস
Re: Set of Natural Numbers
I think second problem is really a bit hard problem :-/
Here's the third problem ( which i couldn't solve yet)
Is it true that for all $c,d \in \mathbb{N}$, $\lim_{n\to\infty}\frac{ \{ n \} }{ \{ c n + d \} } = \frac{1}{c}$ ?
Here's the third problem ( which i couldn't solve yet)
Is it true that for all $c,d \in \mathbb{N}$, $\lim_{n\to\infty}\frac{ \{ n \} }{ \{ c n + d \} } = \frac{1}{c}$ ?
ধনঞ্জয় বিশ্বাস
Re: Set of Natural Numbers
Hm, but now I thought to show $ \frac{ \{ n \} }{ \{ a n + b \} }$ increases as $n$ increases might work and this process might work by increasing $n$ over some gaps like jumping from $n$ to $kn$...
Edited.Do you understand now?
Edited.Do you understand now?
Last edited by *Mahi* on Thu Sep 08, 2011 6:22 pm, edited 1 time in total.
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: Set of Natural Numbers
I can't understand*Mahi* wrote:Hm, but now I thought this would work, to show $ \frac{ \{ n \} }{ \{ a n + b \} }$ increases as $n$ increases, and this might work by increasing $n$ over some gaps...
ধনঞ্জয় বিশ্বাস