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 January 21st, 2017, 12:26 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 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: USA Posts: 2,424 Thanks: 1311 suppose you have two numbers $a,b \in \mathbb{R}$ consider the sequence $s_k=a + \dfrac{b-a}{k},~k=1,2,3,4, \dots$ $s_1=b$ $\displaystyle{\lim_{k\to \infty}}~s_k=a$ $a < s_k \leq b$ and $s_k$ is an infinite sequence January 21st, 2017, 01:34 AM #3 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 thanks. January 21st, 2017, 02:23 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,613 Thanks: 2071 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: 7,649 Thanks: 2630 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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post kustrle Number Theory 4 October 18th, 2015 12:29 PM johnr Number Theory 6 March 3rd, 2015 02:17 PM xianghu21 Applied Math 0 March 24th, 2010 09:18 AM Infinity Number Theory 13 July 21st, 2007 08:35 PM Infinity Applied Math 4 July 3rd, 2007 06:19 PM

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