January 21st, 2017, 01:26 AM  #1 
Senior Member Joined: Nov 2011 Posts: 175 Thanks: 2  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: 1,763 Thanks: 905 
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: 175 Thanks: 2 
thanks.

January 21st, 2017, 03:23 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,594 Thanks: 1492 
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: 7,151 Thanks: 2387 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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