how to prove that there exist infinitely many rationals between any two irrationals?


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Question Number 12405 by Ms.Ramanujan last updated on 21/Apr/17

$h o w t o p r o v e t h a t t h e r e e x i s t i n f i n i t e l y m a n y r a t i o n a l s b e t w e e n a n y t w o i r r a t i o n a l s ?$
Commented by FilupS last updated on 21/Apr/17
$a<x<b let x= (b/(na)) a,b,n∈Z there are an infinte number of rationals between a and b let (b/a)∈R\Q, n∈R x=(b/(na))∈Q e.g. x=(1/(k((1/π))×π)) k∈Q or x=((π+1)/(nπ)) ⇒ n=k(1+(1/π)) ∴∀a∀b∈{R\Q}∃{x}∈(a,b):x∈Q∧∣{x}∣=∞$
$a < x < b$ $let x = \frac{b}{n a} a, b, n \in Z$ $there are an infinte number of rationals$ $between a a n d b$ $let \frac{b}{a} \in R ∖ Q, n \in R$ $x = \frac{b}{n a} \in Q e . g . x = \frac{1}{k (\frac{1}{π}) \times π} k \in Q$ $or x = \frac{π + 1}{n π} \Rightarrow n = k (1 + \frac{1}{π})$ $∴ \forall a \forall b \in {R ∖ Q} \exists {x} \in (a, b) : x \in Q \land ∣ {x} ∣= \infty$
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