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Question Number 12405 by Ms.Ramanujan last updated on 21/Apr/17
how to prove that there exist infinitely many rationals between any two irrationals?
$${how}\:{to}\:{prove}\:{that}\:{there}\:{exist}\:{infinitely}\:{many}\:{rationals}\:{between}\:{any}\:{two}\:{irrationals}? \\ $$$$ \\ $$
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}∣=∞
$${a}<{x}<{b} \\ $$$$\: \\ $$$$\mathrm{let}\:{x}=\:\frac{{b}}{{na}}\:\:\:\:{a},{b},{n}\in\mathbb{Z} \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{an}\:\mathrm{infinte}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rationals} \\ $$$$\mathrm{between}\:{a}\:{and}\:{b} \\ $$$$\: \\ $$$$\mathrm{let}\:\frac{{b}}{{a}}\in\mathbb{R}\backslash\mathbb{Q},\:{n}\in\mathbb{R} \\ $$$${x}=\frac{{b}}{{na}}\in\mathbb{Q}\:\:\:\:\:\mathrm{e}.\mathrm{g}.\:\:{x}=\frac{\mathrm{1}}{{k}\left(\frac{\mathrm{1}}{\pi}\right)×\pi}\:\:\:\:\:\:\:{k}\in\mathbb{Q} \\ $$$$\:\:\:\:\:\:\mathrm{or}\:\:{x}=\frac{\pi+\mathrm{1}}{{n}\pi}\:\Rightarrow\:{n}={k}\left(\mathrm{1}+\frac{\mathrm{1}}{\pi}\right) \\ $$$$\: \\ $$$$\therefore\forall{a}\forall{b}\in\left\{\mathbb{R}\backslash\mathbb{Q}\right\}\exists\left\{{x}\right\}\in\left({a},{b}\right):{x}\in\mathbb{Q}\wedge\mid\left\{{x}\right\}\mid=\infty \\ $$

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