January 21st, 2017, 01:26 AM  #1 
Senior Member Joined: Nov 2011 Posts: 248 Thanks: 3  infinite numbers
How I prove that between any two numbers there are infinite numbers?

January 21st, 2017, 02:15 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,325 Thanks: 1234 
suppose you have two numbers $a,b \in \mathbb{R}$ consider the sequence $s_k=a + \dfrac{ba}{k},~k=1,2,3,4, \dots$ $s_1=b$ $\displaystyle{\lim_{k\to \infty}}~s_k=a$ $a < s_k \leq b$ and $s_k$ is an infinite sequence 
January 21st, 2017, 02:34 AM  #3 
Senior Member Joined: Nov 2011 Posts: 248 Thanks: 3 
thanks.

January 21st, 2017, 03:23 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,310 Thanks: 1981 
You must, of course, be talking about distinct reals (else there would be no reals between them), and not requiring the numbers between them to be integers.

January 21st, 2017, 05:35 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra 
I think that Romsek's answer is overly complicated. Perhaps it's just because he needlessly mentions a limit. Here's a different construction. Given $a_0 \lt a_1$, then \[a_0 \lt a_{n+1}=\frac{a_0+a_n}{2} \lt a_n \] for all $n$ so \[ a_1 \gt a_2 \gt a_3 \gt \ldots \gt a_0 \] 

Tags 
infinite, numbers 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Infinite set of numbers  kustrle  Number Theory  4  October 18th, 2015 01:29 PM 
Infinite Sum of the Inverses of Pronic Numbers  johnr  Number Theory  6  March 3rd, 2015 03:17 PM 
infinite cardinal numbers  xianghu21  Applied Math  0  March 24th, 2010 10:18 AM 
Infinite sum of the reciprocals of the Fibonacci numbers.  Infinity  Number Theory  13  July 21st, 2007 09:35 PM 
Drawing infinite numbers of lines  Infinity  Applied Math  4  July 3rd, 2007 07:19 PM 