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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 = \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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