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January 21st, 2017, 12:26 AM   #1
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infinite numbers

How I prove that between any two numbers there are infinite numbers?
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January 21st, 2017, 01:15 AM   #2
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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
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January 21st, 2017, 01:34 AM   #3
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thanks.
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January 21st, 2017, 02:23 AM   #4
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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.
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January 21st, 2017, 04:35 AM   #5
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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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