January 21st, 2017, 01:26 AM  #1 
Senior Member Joined: Nov 2011 Posts: 116 Thanks: 1  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: Southern California, USA Posts: 1,605 Thanks: 819 
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: 116 Thanks: 1 
thanks.

January 21st, 2017, 03:23 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,157 Thanks: 1422 
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: 7,038 Thanks: 2344 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 \[ a_1 \gt a_2 \gt a_3 \gt \ldots \gt a_0 \] 

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