January 21st, 2017, 12:26 AM  #1 
Member Joined: Nov 2011 Posts: 73 Thanks: 0  infinite numbers
How I prove that between any two numbers there are infinite numbers?

January 21st, 2017, 01:15 AM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,406 Thanks: 714 
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, 01:34 AM  #3 
Member Joined: Nov 2011 Posts: 73 Thanks: 0 
thanks.

January 21st, 2017, 02:23 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 17,914 Thanks: 1382 
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, 04:35 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,937 Thanks: 2265 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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