Question

Prove that there are infinite rational numbers between any two rational number by different methods.​

Answer 1

Answer:

First proof

First, pick a positive irrational number of your choice. Let us call it γγ.

Now, let aa and bb be two rational numbers, with a<ba<b. I will exhibit infinitely many irrational numbers between aa and bb.

Let ϵ=b−a>0ϵ=b−a>0. Since the sequence γ,γ/2,γ/3,γ/4,⋅γ,γ/2,γ/3,γ/4,⋅ converges to 0, there is a (large enough) choice of natural number nn, such that γ/n<ϵγ/n<ϵ. Now a+γna+γn, a+γn+1a+γn+1, a+γn+2a+γn+2, etc. are infinitely many irrationals between aa and bb.

Second proof

Let aa and bb be two rational numbers, with a<b,a<b, such that there are only finitely many irrational numbers in the closed interval [a,b][a,b]. By definition,[a,b][a,b] is the collection of all real numbers xx, where a≤x≤ba≤x≤b. Then the set In=[a+n(b−a

hope it's helpful