Prove that if x and y

ban1ka1u

ban1ka1u

Answered question

2022-07-02

Prove that if x and y are irrational numbers, there exists an irrational number z such that y < z < x.

Answer & Explanation

furniranizq

furniranizq

Beginner2022-07-03Added 20 answers

x y > 0 so ( x y ) 2 > 0. By the Archimedean property, there is a positive integer m such that 1 m < ( x y ) 2 . By increasing m if necessary, you can assume that m = 2 n 2 so that 1 2 n 2 < ( x y ) 2 . This means
1 2 n < x y 1 + 2 n y < 2 n x .
Now let k be the largest integer that is less than or equal to 2 n y. This implies
1 + k 1 + 2 n y < 2 n x .
But we also have 2 n y < k + 1, so
2 n y < k + 1 < 2 n x y < k + 1 2 n < x .

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